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3.1
1.

You can test if the number is divisible by 9. Add the digits together. If that sum is divisible by 9, then so is the original number.

5 + 4 = 9, so 54 is divisible by 9.

Yes. When 54 is divided by 9, the result is 6 with no remainder. Also, 54 can be written as the product of 9 and 6.
3.2
1.
The last digit is 0, so 45,730 is divisible by 5, since the rule states that if the last digit is 0 or 5, the original number is divisible by 5.
3.3
1.
The sum of the digits is 32. Since 32 is not divisible by 9, neither is 342,887.
3.4
1.
The last digit is even, so 2 divides 43,568. The sum of the digits is 26. Since 26 is not divisible by 3, neither is 43,568. The rule for divisibility by 6 is that the number be divisible by both 2 and 3. Since 43,568 is not divisible by 3, it is not divisible by 6.
3.5
1.
Since the last digit of 87,762 is not 0, it is not divisible by 10.
3.6
1.
The number formed by the last two digits of 43,568 is 68 and 68 is divisible by 4. Since the number formed by the last two digits of 43,568 is divisible by 4, so is 43,568.
3.7
1.

You only need to check numbers up to the square root of 1,429, which is approximately 37.80.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is not even.

3: Add the digits of the number together. If that sum is divisible by 3, then so is the original number.

1 + 4 + 2 + 9 = 16

16 is not divisible by 3.

5: If the last digit is 5 or 0, then the original number is divisible by 5.

The last digit is not 5 or 0, so the number is not divisible by 5.

Step 2:

Use a calculator to test the primes up to 37.

7, 11, 13, 17, 19, 23, 29, 31, 37

For instance, 1,429 ÷ 7 ≈ 178.4285714.

None of these primes result in an integer quotient. They all have a decimal part.

The conclusion is that 1,429 is a prime number.

Yes, 1,429 is prime.
3.8
1.

You only need to check numbers up to the square root of 859, which is approximately 29.308.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is not even.

3: Add the digits of the number together. If that sum is divisible by 3, then so is the original number.

8 + 5 = 13

13 is not divisible by 3.

5: If the last digit is 5 or 0, then the original number is divisible by 5.

The last digit is not 5 or 0, so the number is not divisible by 5.

Step 2:

Use a calculator to test the primes up to 29.

7, 11, 13, 17, 19, 23, 29

For instance, 859 ÷ 7 ≈ 122.7142857.

None of these primes result in an integer quotient. They all have a decimal part.

The conclusion is that 859 is a prime number.

Yes, 859 is a prime number.
3.9
1.

You only need to check numbers up to the square root of 5,067,322, which is approximately 2251.071301.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is even.

5,067,322 = 2 × 2,533,661

The conclusion is that 5,067,322 is a composite number.

5,067,322 is a composite number.
3.10
1.

You only need to check numbers up to the square root of 1,477, which is approximately 38.4317577.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is not even.

3: Add the digits of the number together. If that sum is divisible by 3, then so is the original number.

1 + 4 + 7 + 7 = 19

19 is not divisible by 3.

5: If the last digit is 5 or 0, then the original number is divisible by 5.

The last digit is not 5 or 0, so the number is not divisible by 5.

Step 2:

Use a calculator to test the primes up to 38.

7, 11, 13, 17, 19, 23, 29, 31, 37

For instance, 1,477 ÷ 7 = 211.

7 is an integer factor, so the conclusion is that 1,477 is a composite number.

No, 1,477 is composite.
3.11
1.

Use the divisibility rules for the primes, starting from the smallest: 2, 3, 5, 7, 11, 13, 17, 19.

Because 90 is even, you know it is divisible by 2.

90 = 2 × 45

Because 45 ends in 5, you know it is divisible by 5.

90 = 2 × 5 × 9

90 = 2 × 5 × 3 × 3

90 = 2 × 3 2 × 5

90 = 2 × 3 2 × 5
3.12
1.

You know 85 has a factor of 5 because it ends in 5.

85 = 5 × 17

Because 17 is a prime number, you are done.

85 = 5 × 17
3.13
1.

Because 280 ends in 0, you know it has a factor of 10, which is 2 times 5. Factor out 2 and 5.

280 = 2 × 5 × 28

28 is divisible by 2, so factor out 2 again.

280 = 2 × 2 × 5 × 14

14 is even, so factor out 2 again.

280 = 2 × 2 × 2 × 5 × 7

Or you can write the factorization: 280 = 2 3 × 5 × 7 .

2 3 × 5 × 7
3.14
1.

Because 180 ends in 0, you know it has a factor of 10, which is 2 times 5. Factor out 2 and 5.

180 = 2 × 5 × 18

18 is divisible by 2, so factor out 2 again.

180 = 2 × 2 × 5 × 9

Factor 9.

180 = 2 × 2 × 5 × 3 × 3

180 has three prime factors (2, 3, and 5).

The number 180 has three prime factors.
3.15
1.
Number Factors
270 2 3 3 3 5  
99   3 3     11
Common   3 3      
Greatest common divisor is the product of the common factors: 3 × 3 = 9 .
The GCD is 9.
3.16
1.
Number Factors
36 2 2           3 3
128 2 2 2 2 2 2 2    
Common 2 2              
Greatest common divisor is the product of the common factors: 2 × 2 = 4 .
The GCD of 36 and 128 is 4.
3.17
1.
Number Factors
120 2 2 2 3 5  
200 2 2 2 5 5
Common 2 2 2   5  
Greatest common divisor is the product of the common factors: 2 × 2 × 2 × 5 = 40
40
3.18
1.

You need the GCD of the width and length: 400 cm by 540 cm

Number Factors
400 2 2 2 2       5 5
540 2 2 3 3 3 5  
Common 2 2           5  

Greatest common divisor = 2 × 2 × 5 = 20   c m

The largest square bricks that can be used are 20 cm by 20 cm.

The largest square bricks that can be used are 20 cm by 20 cm.
3.19
1.

Find the GCD of 21, 35, and 28.

Number Factors        
21     3   7
35     5 7
28 2 2     7
Common         7

Greatest common divisor = 7

The largest team size is 7 students.

The largest team size that can be formed is 7 students.
3.20
1.

Multiples of 12: 12, 24, 36, 48, 60

Multiples of 15: 15, 30, 45, 60

The first number common to both lists is 60.

The LCM is 60.

60
3.21
1.

Find the prime factorization of each number.

20 = 2 2 × 5

28 = 2 2 × 7

Make a table of each prime and the largest exponent for each prime.

Prime 2 5 7
Exponent 2 1 1

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is 2 2 × 5 × 7 = 140 .

140
3.22
1.

Find the prime factorization of each number.

150 = 2 × 3 × 5 2

240 = 2 4 × 3 × 5

462 = 2 × 3 × 7 × 11

Make a table of each prime and the largest exponent for each prime.

Prime 2 3 5 7 11
Exponent 4 1 2 1 1

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is 2 4 × 3 × 5 2 × 7 × 11 = 92 , 400 .

92,400
3.23
1.

Using lists:

List multiples of each number until you find a common number.

18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270, 288, 306, 324, 342, 360

24: 24, 48, 72, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360

40: 40, 80, 120, 160, 200, 240, 280, 320, 360

The first common number is 360. The LCM is 360.

Using prime factorization:

Find the prime factorization of each number.

18 = 2 × 3 2

24 = 2 3 × 3

40 = 2 3 × 5

Make a table of each prime and the largest exponent for each prime.

Prime 2 3 5
Exponent 3 2 1

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is 2 3 × 3 2 × 5 = 360 .

360
3.24
1.

Find the LCM of 255 and 4,330.

Find the prime factorization of each number.

255 = 3 × 5 × 17

4 , 330 = 2 × 5 × 433

Make a table of each prime and the largest exponent for each prime.

Prime 2 3 5 17 433
Exponent 1 1 1 1 1

The LCM is the product of each prime raised to the powers identified in the table.

The LCM i 2 × 3 × 5 × 17 × 433 = 220 , 830 .

The sun, Venus, and Jupiter will align in 220,830 days.

The sun, Venus, and Jupiter will line up again in 220,830 days.
3.25
1.

Find the LCM of 130 and 900.

Find the prime factorization of each number.

255 = 2 × 5 × 13

900 = 2 2 × 3 2 × 5 2

Make a table of each prime and the largest exponent for each prime.

Prime 2 3 5 13
Exponent 2 2 2 1

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is 2 2 × 3 2 × 5 2 × 13 = 11 , 700 .

The prize winner is the one who submits the 11,700th submission.

The first person to receive both giveaways would be the person who submits the 11,700th submission.
3.26
1.
–214 is an integer, as it is the negative of a counting number.
integer
2.
38 / 11 is not an integer. It is approximately 3.45, which is between 3 and 4, so it is between consecutive integers.
not an integer
3.
The square root of 90 is approximately 9.4868, which is between 9 and 10. This is not an integer.
not an integer
4.
The square root of 121 is 11. This is an integer.
integer
5.
Dividing 420 by 35 results in 12, which is an integer. This is an integer.
integer
3.27
1.
On a number line marked with tick marks for every integer, count 10 tick marks in the negative direction. Place a solid dot at that point.

A number line ranges from negative 10 to 10, in increments of 1. A point is marked at negative 10 and it is labeled x equals negative 10.
2.
On a number line marked with tick marks for every integer, count four tick marks in the positive direction. Place a solid dot at that point.

A number line ranges from negative 7 to 7, in increments of 1. A point is marked at 4 and it is labeled x equals 4.
3.
On a number line marked with tick marks for every integer, place a solid dot at zero.

A number line ranges from negative 7 to 7, in increments of 1. A point is marked at 0 and it is labeled x equals 0.
3.28
1.

A number line ranges from negative 42 to 30, in increments of 6. Two points are marked at negative 38 and 27.
27 > 38 and 38 < 27

Consider a number line with one tick mark for every integer. To graph –38, you move 38 tick marks to the left of zero. To graph 27, you move 27 tick marks to the right of zero.

–38 is farther left than 27, so –38 is less than 27.

38 < 27

27 > 38

3.29
1.

Consider a number line with one tick mark for every integer. To graph –213, you move 213 tick marks to the left of zero. To graph –63, you move 63 tick marks to the left of zero.

–213 is farther left than –63, so –213 is less than –63.

213 < 63

63 > 213


A 
number line ranges from negative 220 to negative 60, in increments of 10. Two points are marked at negative 213 and negative 63. 63 > 213 and 213 < 63
3.30
1.
101 is larger. 101 > 98 and 98 < 101 .
101 is to the right of 98 on a number line. Thus, 101 > 98 and 98 < 101 .
3.31
1.
38

Because the value inside the absolute value is positive, the absolute value is just the number itself.

| 38 | = 38

3.32
1.

Because the value inside the absolute value bars is negative, the absolute value removes the negative sign.

|–81| = 81

81
3.33
1.
Use your calculator to find that (–18) + 11 is –7. Because you are adding numbers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value. Because |–18| > |11|, the sign of the answer matches the sign of –18.
−7. Since |−18| > |11|, the answer matches the sign of −18.
3.34
1.

Use your calculator to find that 38 – 100 is –62.

Because 100 is larger than 38, you expect the difference to be negative.

−62. Since a larger positive number was subtracted from a smaller positive number, a negative result was expected.
3.35
1.
Use your calculator to find that 45 – (–26) is 71. Subtracting a negative number is the same as adding a positive number. This is the same as adding 45 + 26 to get 71.
71. Subtracting a negative number is the same as adding a positive number.
3.36
1.

Use your calculator to find that 19 + (–36) is –17.

Because you are adding numbers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value. Because |–36| > |19|, the sign of the answer matches the sign of –36.

−17. Since |19| < |−36|, the sign of the answer matches the sign of −36, which is negative.
3.37
1.
$89

Chanel owes Chrisian money, so represent $180 as a positive number. Christian owes Jeff money, so subtract 91 from 180.

180 91 = 89

Christian’s net worth is $89.

3.38
1.
2,106. Since both numbers are positive, the product is positive.

81 × 26   i s   2 , 106 .

Because 81 and 26 are both positive, the product is positive.

3.39
1.
−234. Since the numbers have opposite signs, the product is negative.

( 18 ) × 13   i s   234 .

Because –18 and 13 have opposite signs, the product is negative.

3.40
1.
−29. The numbers have opposite signs, so the division will result in a negative number.

( 116 ) ÷ 4   i s   29 .

Because –116 and 4 have opposite signs, the quotient is negative.

3.41
1.
7. Since the signs of the numbers match, the division results in a positive number.

( 77 ) ÷ ( 11 )   i s   7 .

Because –77 and –11 have the same sign, the quotient is positive.

3.42
1.
The average daily balance was $529.

Add the seven account balances, then divide by 7.

Sum: 1250 + 673 + ( 1500 ) + 1000 + 785 + 785 + 710

Divide the sum by 7:

3 , 703 ÷ 7 = 529

The average balance is $529.

3.43
1.

43 + 18 × 15

Use PEMDAS. The highest order operation in this expression is multiplication.

Multiply. 43 + 270
Add. 313
313
3.44
1.

60 2 / 50

Use PEMDAS.

Calculate the exponent. 3600 / 50
Divide. 72
72
3.45
1.

3 × 6 3 18

Use PEMDAS.

Calculate the exponent. 3 × 216 18
Multiply. 648 18
Subtract. 630
630
3.46
1.

13 + 3 4 + 8 / 6 + 5 2 3 × 4

Use PEMDAS.

Calculate the exponents. 13 + 81 × 8 / 6 × 25 3 × 4
Multiply times 8. 13 + 648 / 6 × 25 3 × 4
Divide by 6. 13 + 108 × 25 3 × 4
Multiply by 25. 13 + 2 , 700 3 × 4
Multiply by 4. 13 + 2 , 700 12
Add. 2 , 713 12
Subtract. 2 , 701
2,701
3.47
1.

12 / ( 4 ) + 8 × 9 / 12 × 2 3 24 × 25 / 10

Use PEMDAS.

Calculate the exponent. 12 / ( 4 ) + 8 × 9 / 12 × 8 24 × 25 / 10
Divide by –4. 3 + 8 × 9 / 12 × 8 24 × 25 / 10
Multiply by 9. 3 + 72 / 12 × 8 24 × 25 / 10
Divide by 12. 3 + 6 × 8 24 × 25 / 10
Multiply by 8. 3 + 48 24 × 25 / 10
Multiply by 25. 3 + 48 600 / 10
Divide by 10. 3 + 48 60
Add. 45 60
Subtract. 15
−15
3.48
1.

3 × 4 3 × 7 + 24 / 6 × 7 2 9 / 3 × 8

Use PEMDAS.

Calculate the exponents. 3 × 64 × 7 + 24 / 6 × 49 9 / 3 × 8
Multiply times 64. 192 × 7 + 24 / 6 × 49 9 / 3 × 8
Multiply times 7. 1 , 344 + 24 / 6 × 49 9 / 3 × 8
Divide by 6. 1 , 344 + 4 × 49 9 / 3 × 8
Multiply by 49. 1 , 344 + 196 9 / 3 × 8
Divide by 3. 1 , 344 + 196 3 × 8
Multiply by 8. 1 , 344 + 196 24
Add. 1 , 540 24
Subtract. 1 , 516
1,516
3.49
1.

8 ( 25 2 2 ) / 7

Use PEMDAS.

Evaluate the operations inside the parentheses first.

Calculate the exponent. 8 ( 25 4 ) / 7
Subtract inside the parentheses. 8 ( 21 ) / 7
Divide by 7. 8 3
Subtract. 5
5
3.50
1.

( 8 6 ) 2 × 100 ( ( 48 / 6 3 ) 2 4 × 7 )

Use PEMDAS.

Evaluate the operations inside the parentheses first.

Subtract inside the first parentheses. ( 2 ) 2 × 100 ( ( 48 / 6 3 ) 2 4 × 7 )
Divide by 6 inside the inner parentheses. ( 2 ) 2 × 100 ( ( 8 3 ) 2 4 × 7 )
Subtract inside the inner parentheses. ( 2 ) 2 × 100 ( ( 5 ) 2 4 × 7 )
Evaluate the exponent inside the second parentheses. ( 2 ) 2 × 100 ( 25 4 × 7 )
Multiply by 7 inside the second parentheses. ( 2 ) 2 × 100 ( 25 28 )
Subtract inside the second parentheses. ( 2 ) 2 × 100 ( 3 )
Evaluate the exponent. 4 × 100 ( 3 )
Multiply by 100. 400 ( 3 )
Subtract. 403
403
3.51
1.

94 is not a perfect square.

The prime factorization of 94 is 2 × 47 . None of the factors is a pair.

94 is not a perfect square.
2.
441 is a perfect square. 21 × 21 = 441
441 is a perfect square.
3.52
1.

Not rational

13 3.60555.... . There is no repeating pattern, so this is not a rational number.

not a rational number
2.
This is a rational number because it is a repeating decimal.
rational number
3.
This is a rational number because it is the ratio of two integers.
rational number
4.
This is a rational number. A mixed number can be expressed as the ratio of two integers.
rational number
5.
This is a rational number. A decimal that terminates can be expressed as a ratio of two integers.
rational number
3.53
1.

Two fractions, a b and c d are equivalent if a × d = b × c .

a × d = 8 × 26 = 208

b × c = 14 × 12 = 168

Because the products are not the same, the fractions are not equivalent.

a × b = 8 × 26 = 208 and b × c = 14 × 12 = 168 . The fractions are not equivalent.
3.54
1.
Write the fraction in factored form. 2 × 2 × 3 × 3 × 7 2 × 2 × 2 × 3 × 5 × 7
Divide out the common factors. 2 × 2 × 3 × 3 × 7 2 × 2 × 2 × 3 × 5 × 7
Simplify. 3 10
3 10
3.55
1.

Enter the fractions in Desmos (desmos.com) using “/” for the fraction bar.

What you enter: 124 / 297

You see a fraction in the workspace and a decimal in the answer area.

Arrow out of the fraction and press + .

Enter the second fraction: 3 / 125

The answer area shows 0.4415084175.

Click on the “convert to fraction button.” It looks like .

You will see the final answer, 16 , 391 37 , 125 .

16 , 391 37 , 125
3.56
1.

38 73 + 7 73

When two fractions have the same denominator, add the numerators.

38 + 7 73

45 73

There are no common factors to divide out.

45 73
3.57
1.

21 40 8 40

When two fractions have the same denominator, subtract the numerators.

21 8 40

13 40

There are no common factors to divide out.

13 40
3.58
1.

4 9 + 7 12

Find the LCM of 9 and 12. (You can use Desmos to do this.)

Number Factors
9 3 3
12 2 2 3
LCM 2 2 3 3

L C M = 2 × 2 × 3 × 3 = 36

Rewrite the fractions with 36 as the denominator.

4 × 4 9 × 4 + 7 × 3 12 × 3

16 36 + 21 36

Now, you can add the fractions by adding the numerators.

37 36

There are no common factors to divide out.

Alternative answer: 1 1 36

37 36
3.59
1.

10 99 17 300

Find the LCM of 99 and 300. (You can use Desmos to do this.)

Number Factors
99 3 3 11
300 2 2 3 5 5
LCM 2 2 3 3 5 5 11

L C M = 2 × 2 × 3 × 3 × 5 × 5 × 11 = 9 , 900

Rewrite the fractions with 9,900 as the denominator.

10 × 100 99 × 100 17 × 33 300 × 33

1 , 000 9 , 900 561 9 , 900

Now, you can subtract the fractions by subtracting the numerators.

1 , 000 561 9 , 900

439 9 , 900

There are no common factors to divide out.

439 9 , 900
3.60
1.

95 ÷ 26 is 3 with a remainder. What is that remainder?

26 × 3 = 78

95 78 = 17

The remainder is 17.

So, write 95 26 as 3 17 26 .

3 17 26
3.61
1.

9 5 14

Multiply the integer part by the denominator. 9 × 14 = 126
Add the product to the numerator. 126 + 5 = 131
Write the sum divided by the denominator. 131 14
131 14
3.62
1.

The third decimal digit is 8. The next digit is 2, so round down.

5.108

5.108
3.63
1.

The second decimal digit is 2. The next digit is 9, so round up. Change the 2 to 3.

18.63

18.63
3.64
1.

Use a calculator to divide 48 by 30 to get 1.6.

Or because you can simplify to have 10 in the denominator, you can also find the decimal this way.

48 30 = 3 × 16 3 × 10

16 10 = 1.6

1.6
3.65
1.

There are five decimal places, so raise 10 to the fifth power.

10 5 = 100 , 000

17.03347 × 100 , 000 1 × 100 , 000 = 1 , 703 , 347 100 , 000

1 , 703 , 347 100 , 000
3.66
1.

45 88 × 28 75

Multiply the numerators and place that in the numerator.

Multiply the denominators and place that in the denominator.

45 × 28 88 × 75

1 , 260 6 , 600

Simplify. The GCF of 1,260 and 6,600 is 60.

60 × 21 60 × 110

21 110

21 110
3.67
1.

45.63 ÷ 17.13 is approximately 2.663747 . . . .

The third decimal place is 3. The next digit is 7, so round up. Change the 3 to 4.

2.664

2.664
3.68
1.

46 175 ÷ 69 285

Multiply the first fraction by the reciprocal of the second fraction.

46 175 × 285 69 = 13 , 110 12 , 075

Simplify.

38 × 345 35 × 345

38 35

38 35
2.

3 40 ÷ 42 55

Multiply the first fraction by the reciprocal of the second fraction.

3 40 × 55 42 = 165 1 , 680

Simplify.

11 × 15 112 × 15

11 112

11 112
3.69
1.

( 3 16 + 7 16 ) 2 + 1 5 ÷ 3 10

Use PEMDAS.

Add the fractions inside the parentheses. ( 10 16 ) 2 + 1 5 ÷ 3 10
Simplify. ( 5 8 ) 2 + 1 5 ÷ 3 10
Evaluate the exponent. 25 64 + 1 5 ÷ 3 10
Divide. Multiply by the reciprocal. 25 64 + 1 5 × 10 3
25 64 + 10 15
25 64 + 2 3
Add. The LCM is 192. 25 × 3 64 × 3 + 2 × 64 3 × 64
75 192 + 128 192
203 192
Alternate correct answer: 1 11 192
203 192
3.70
1.

( 3 5 + 2 ) × ( 4 5 1 2 ) 2 ÷ 11 15

Use PEMDAS.

Add the numbers inside the parentheses. ( 3 5 + 10 5 ) × ( 4 5 1 2 ) 2 ÷ 11 15
13 5 × ( 4 5 1 2 ) 2 ÷ 11 15
Subtract the fractions inside the parentheses. 13 5 × ( 8 10 5 10 ) 2 ÷ 11 15
Calculate the exponent. 13 5 × ( 3 10 ) 2 ÷ 11 15
Multiply. 117 500 ÷ 11 15
Divide. Multiply by the reciprocal. 117 100 × 5 × 3 × 5 11
117 500 × 15 11
351 1 , 100

Optional decimal form: 0.31 90 ¯

351 1 , 100 , or in decimal form, 0.319 09 ¯
3.71
1.

The density property process:

Step 1: Add the two rational numbers.

27 13 + 21 10 = 27 × 10 13 × 10 + 21 × 13 10 × 13

270 130 + 273 130

543 130

Step 2: Divide that result by 2. Dividing by 2 is the same as multiplying by one-half.

543 130 × 1 2 = 543 260

There are other numbers, but this is the one found using this process.

The process used above yields 543 260 .
3.72
1.

Math: half the t i m e = 1 2 × 10 = 5 h o u r s on math

History: a quarter of the t i m e = 1 4 × 10 = 2.5 h o u r s on history

Writing: an eighth of the t i m e = 1 8 × 10 = 1.25 h o u r s on writing

Physics: an eighth of the t i m e = 1 8 × 10 = 1.25 h o u r s on physics


Studying math: 5 hours
Studying history: 2.5 hours
Studying writing: 1.25 hours
Studying physics: 1.25 hours
3.73
1.

30% of 2,400 calories

30 100 × 2 , 400 = 720

Callum should consume 720 calories of protein.

720 calories of protein
3.74
1.
200 m i l e s ( 1.60934 k m 1 m i l e ) 321.868 k m
321.868 km
3.75
1.
25 q u a r t s ( 3 .785 l i t e r s 4 q u a r t s ) 23.656 l i t e r s
23.656 liters
3.76
1.

A percent is a specific rational number and is literally per 100. n percent, denoted n % , is the fraction n 100 .

4 % = 4 100

Alternate correct answer: 1 25

4 100
2.

A percent is a specific rational number and is literally per 100. n percent, denoted n % , is the fraction n 100 .

50 % = 50 100

Alternate correct answer: 1 2

50 100
3.77
1.

A percent is a specific rational number and is literally per 100. n percent, denoted n % , is the fraction n 100 .

14 % = 14 100

In decimal form: 0.14

0.14
2.

A percent is a specific rational number and is literally per 100. n percent, denoted n % , is the fraction n 100 .

7 % = 7 100

In decimal form: 0.07

0.07
3.78
1.

25% of 1,200

Rewrite “of” as multiplication. 25 100 × 1 , 200
Simplify. 25 × 12 × 100 100
300
300
2.

53% of 1,588

Rewrite “of” as multiplication. 53 100 × 1 , 588
Use your calculator. 841.64
841.64
3.79
1.

25% of the total is 30.

Rewrite “of” as multiplication. 25 100 × 30
Use your calculator. 120
120
2.

45% of the total is 360.

n % of x items is n 100 × x .

Rewrite “of” as multiplication and “is” as equals. 45 100 × t o t a l = 360
Multiply both sides by 100 45 . 100 45 ( 45 100 × t o t a l ) = ( 360 ) ( 100 45 )
100 45 ( 45 100 × t o t a l ) = ( 360 ) ( 100 45 )
t o t a l = 800
800
3.80
1.

n percent, denoted n % , is the fraction n 100 .

70 is what percentage of 1,000?

Rewrite “is” as equals and “percentage of” as n 100 × 1 , 000.

70 = n 100 × 1 , 000

70 = 10 n Divide by 10.

7 = n

The answer is 7%.

7%
2.

n percent, denoted n % , is the fraction n 100 .

425 is what percentage of 500?

Rewrite “is” as equals and “percentage of” as n 100 × 500.

425 = n 100 × 500

425 = 5 n Divide by 5.

85 = n

The answer is 85%.

85%
3.81
1.

What is 20% of 2,200?

n percent, denoted n % , is the fraction n 100 .

Rewrite “of” as multiplication. 20 100 × 2 , 200
Use your calculator. 440

Rily should eat 440 calories of protein.

440 calories of protein
3.82
1.

n percent, denoted n % , is the fraction n 100 .

54 is what percentage of 450?

Rewrite “is” as equals and “percentage of” as n 100 × 500.

54 = n 100 × 450

54 = 4.5 n Divide by 4.5.

12 = n

The answer is 12% of registered voters voted in the primaries.

12% of registered voters in the small town voted in the primaries.
3.83
1.

40% of the total is $30.

n % of x items is n 100 × x .

Rewrite “of” as multiplication and “is” as equals. 40 100 × t o t a l = 30
Multiply both sides by 100 40 . 100 40 ( 40 100 × t o t a l ) = ( 30 ) ( 100 40 )
100 40 ( 40 100 × t o t a l ) = ( 30 ) ( 100 40 )
t o t a l = $ 75

The original price was $75.

We want the original price of the item, which is the total. We know the percent, 40, and the percentage of the total, $30. To find the original cost, use 100 × x n , with x = 30 and n  =  40 . Calculating with those values yields 100 × 30 40 = 75 . So, the original was $75.
3.84
1.
36 is a perfect square as it equals 6 2 .
perfect square
2.
27 is not a perfect square. It is a perfect cube as its prime factorization is 3 3 .
not a perfect square
3.
9 49 = ( 3 7 ) 2

This is a perfect square.

perfect square
4.

12 221 = 3 × 4 13 × 17

This is not a perfect square as you do not have pairs of prime factors.

not a perfect square
3.85
1.
rational

225 = 15

This is not irrational because you can write 15 as the ratio of two integers.

2.
irrational
This equals a decimal that does not repeat, so this is an irrational number.
3.
irrational
This equals a decimal that does not repeat, so this is an irrational number.
4.
irrational
This equals a decimal that does not repeat, so this is an irrational number.
3.86
1.

550 = 25 22 = 5 22

The rational part is 5.

The irrational part is 22 .

5 22 . The rational part is 5, and the irrational part is 22 .
3.87
1.

773

773 is a prime number, so this cannot be further simplified.

The rational part is 1. (You can always have a factor of 1.)

The irrational part is 15 .

733 . The rational part is 1, and the irrational part is 733 .
3.88
1.

1 , 815 = 121 15 = 11 15

The rational part is 11.

The irrational part is 15 .

11 15 . The rational part is 11, and the irrational part is 15 .
3.89
1.
18 15

41 15 23 15

Because they have the same irrational part, you can subtract without using a calculator.

The formula uses the distributive property: ( a × x ) ± ( b × x ) = ( a ± b ) × x .

( 41 23 ) × 15 Subtract.

18 15

3.90
1.

4.1 π + 3.2 π

Because they have the same irrational part, you can add without using a calculator.

The formula uses the distributive property: ( a × x ) ± ( b × x ) = ( a ± b ) × x .

( 4.1 + 3.2 ) × π Add.

7.3 π

7.3 π
3.91
1.
These two numbers have different irrational parts, so they cannot be subtracted in their current state. You could use a calculator. You could also modify the first number to make it have the same rational part as the second number.
The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed without a calculator.
3.92
1.

84 132 ÷ ( 14 11 )

Step 1: Divide the rational part.

84 ÷ 14 = 6

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Divide the irrational parts.

132 ÷ 11 = 12

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

12 = 4 3 = 2 3

Step 5: The result is the product of the rational and the irrational parts.

6 ( 2 3 ) = 12 3

12 3
2.

57 792 ÷ ( 25 2 )

Step 1: Divide the rational part.

57 ÷ 25 = 57 25

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Divide the irrational parts.

792 ÷ 2 = 792 2 = 396

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

396 = 36 11 = 6 11

Step 5: The result is the product of the rational and the irrational parts.

57 25 ( 6 11 ) = 342 25 11

342 25 11
3.93
1.

1.2 21 × ( 4.5 14 )

Step 1: Multiply the rational part.

1.2 × 4.5 = 5.4

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Multiply the irrational parts.

21 × 14 = 294

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

294 = 49 6 = 7 6

Step 5: The result is the product of the rational and the irrational parts.

5.4 ( 7 6 ) = 37.8 6

37.8 6
2.

38 π ÷ ( 2 π )

Step 1: Divide the rational part.

38 ÷ 2 = 19

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Divide the irrational parts.

π ÷ π = 1

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

Not necessary.

Step 5: The result is the product of the rational and the irrational parts.

19 × 1 = 19

19
3.94
1.

24 15

Multiply the numerator and denominator by 15 .

24 × 15 15 × 15

Simplify.

24 15 15

Divide out 3.

8 15 5

8 15 5
2.

11 14 6 21

Multiply the numerator and denominator by 21 .

11 14 × 21 6 21 × 21

Simplify.

11 14 × 21 6 × 21

Divide out 3.

11 49 6 6 × 21

11 × 7 6 6 × 7 × 3

11 6 18

11 6 18
3.95
1.

15 5 13

Step 1: Recognize that the denominator is the sum or difference of two numbers, one or both involving square roots. This means the conjugate can be used to remove the square root from the denominator.

Step 2: Multiply the numerator and denominator by the conjugate of the denominator: 5 + 13 .

15 5 13 × 5 + 13 5 + 13

Step 3: In the denominator, remember that a sum times a difference is a difference of squares.

15 5 13 × 5 + 13 5 + 13 = 15 × ( 5 + 13 ) 5 2 ( 13 ) 2 = 15 × ( 5 + 13 ) 25 13 = 15 × ( 5 + 13 ) 12

Step 4: In the numerator, apply the distributive property.

15 × ( 5 + 13 ) 12 = 75 + 15 13 12

Step 5: You can write the answer as two separate numbers by recalling that a b c = a c b c .

75 + 15 13 12 = 75 12 + 15 13 12

Simplify.

25 4 + 5 13 4

Check Your Understanding

25 4 + 5 13 4
3.96
1.
This is a real number. It is a ratio of two integers, making it a rational number. Rational numbers are real numbers.
real
2.
The number i is not a real number. Any number containing it is not a real number.
not real
3.
This number can be represented by a decimal that does not repeat. It is an irrational number, which is a real number.
real
3.97
1.
The only subset this number belongs to is the irrational numbers. In decimal form, it does not repeat.
irrational number
2.
This is an integer and a rational number because it can be written as the ratio of two integers.
integer, rational number
3.
This is a rational number because it can be (and is) written as the ratio of two integers.
rational number
3.98
1.

–4 belongs in the Z area as it is an integer.

13.863 belongs in the Q area as it is a rational number. Terminating decimals can be written as the ratio of two integers.

15 and 871 belong in the N area as they are natural numbers.

3 π and 5 2 belong in the R area as they are real numbers. They are irrational and belong in none of the other areas.

A Venn diagram shows four concentric ovals. The ovals are labeled from inner to outer as follows: N, Z, Q, and R. The oval, N reads, 15 and 871. The oval, Z reads, negative 4. The oval, Q reads, 13.863. The oval, R reads, 5 times square root of 2 and negative 3 pi.

Venn diagram showing ‒4, 13.863, 15, 871, 5 2 , and 3 π
3.99
1.

When using the distributive property, multiplication distributes across addition.

a × ( b + c ) = ( a × b ) + ( a × c )

dstributive property
2.

The additive inverse property tells you that every number plus its negative results in zero.

a + ( a ) = 0

additive inverse property
3.100
1.
9 × 11 = 99 . Using that, the problem can be changed to 99 × 8 . Change to 99 = ( 100 1 ) . Using the distributive property, 99 × 8 = ( 100 1 ) × 8 = 100 × 8 1 × 8 = 800 8 = 792 .
3.101
1.

Determine the remainder when 93 is divided by 12.

93 ÷ 12 = 7 and a remainder. What is that remainder?

12 × 7 = 84

Subtract 84 from 93.

93 84 = 9

The remainder is 9.

The remainder is the solution to 93 mod 12.

93 mod 12 ≡ 9

93 = 9 (mod 12)
2.

Determine the remainder when 387 is divided by 12.

387 ÷ 12 = 32 and a remainder. What is that remainder?

12 × 32 = 384

Subtract 384 from 387.

387 384 = 3

The remainder is 3.

The remainder is the solution to 387 mod 12.

387 mod 12 ≡ 3

387 = 3 (mod 12)
3.102
1.

First, find 43 mod 12.

Determine the remainder when 43 is divided by 12.

43 ÷ 12 = 3 and a remainder. What is that remainder?

12 × 3 = 36

Subtract 36 from 43.

43 36 = 7

The remainder is 7.

The remainder is the solution to 43 mod 12.

43 mod 12 ≡ 7

Now, add 7 hours to 9:00.

The time 7 hours after 9:00 is 4:00.

4:00
3.103
1.

First, find 34 mod 12.

Determine the remainder when 34 is divided by 12.

34 ÷ 12 = 2 and a remainder. What is that remainder?

12 × 2 = 24

Subtract 24 from 34.

34 24 = 10

The remainder is 10.

The remainder is the solution to 34 mod 12.

34 mod 12 ≡ 10

Now, subtract 10 hours from 7:00.

The time 10 hours before 7:00 was 9:00.

9:00
3.104
1.

First, multiply 4 and 19 to get 76.

Now, find 76 mod 12.

Determine the remainder when 76 is divided by 12.

76 ÷ 12 = 6 and a remainder. What is that remainder?

12 × 6 = 72

Subtract 72 from 76.

76 72 = 4

The remainder is 4.

The remainder is the solution to 76 mod 12.

76 mod 12 ≡ 4

The product of 4 and 19 modulo 12 is 4.

4
3.105
1.

You text every three hours for 15 times. You texted for 45 hours.

You need to find 45 modulo 12.

45 ÷ 12 = 3 and a remainder. What is that remainder?

12 × 3 = 36

Subtract 36 from 45.

45 36 = 9

The remainder is 9.

The remainder is the solution to 45 modulo 12.

45 modulo 12 ≡ 9

Nine hours after 8 AM is 5 PM.

5:00
3.106
1.

If you prepared the meal 20 more times and you do it every 5 days, then 100 days have passed.

To find what day it is, you need to find 100 modulo 7.

100 ÷ 7 = 14 and a remainder. What is that remainder?

7 × 14 = 98

Subtract 98 from 100.

100 98 = 2

The remainder is 2.

The remainder is the solution to 100 modulo 7.

100 modulo 7 ≡ 2

Two days after Tuesday is Thursday.

It will be Thursday after you have prepared the meals 20 more times.

Thursday
3.107
1.

Use the product rule: a m × a n = a m + n .

12 13 × 12 8 = 12 13 + 8 = 12 21

12 21
2.
Since the bases are not the same (one is 3, the other 4), this cannot be simplified using the product rule for exponents.
3.108
1.

Use the product rule: a m × a n = a m + n .

b 6 × b 3 = b 6 + 3 = b 9

b 9
3.109
1.

Use the quotient rule: a n a m = a ( n m ) .

b 6 b 4 = b ( 6 4 ) = b 2

b 2
3.110
1.

Use the distributive rule for exponents: ( a × b ) n = a n × b n .

( 2 × 19 ) 14 = 2 14 × 19 14

2 14 × 19 14
3.111
1.

Use the distributive rule for exponents: ( a × b ) n = a n × b n .

( a × b ) 6 = a 6 × b 6

a 6 × b 6
3.112
1.

Use the exponent distributive rule. When you have a fraction, a b , raised to an exponent, n, then ( a b ) n = a n b n .

( 14 5 ) 9 = 14 9 5 9

14 9 5 9
2.

Use the exponent distributive rule. When you have a fraction, a b , raised to an exponent, n, then ( a b ) n = a n b n .

( a 18 ) 5 = a 5 18 5

a 5 18 5
3.113
1.

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents, which is ( a n ) m = a ( n × m ) .

( 11 4 ) 12 = 11 ( 4 × 12 ) = 11 48

11 48
2.

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents, which is ( a n ) m = a ( n × m ) .

( a 7 ) 6 = a ( 7 × 6 ) = a 42

a 42
3.114
1.

Use the negative exponent rule. a n = 1 a n provided that a ≠ 0.

12 3 × 7 5 = 1 12 3 × 7 5 = 7 5 12 3

7 5 12 3
2.

Use the negative exponent rule. a n = 1 a n provided that a ≠ 0.

c 7 × 5 3 = 1 c 7 × 5 3 = 5 3 c 7

5 3 c 7
3.115
1.
6 3 × 13 8

Use the negative exponent rule. a n = 1 a n provided that a ≠ 0.

The negative exponent rule lets you say 1 13 8 is equal to 13 8 .

6 3 13 8 = 6 3 × 13 8

2.
c 5 × 2 9

Use the negative exponent rule. a n = 1 a n provided that a ≠ 0.

The negative exponent rule lets you say 1 2 9 is equal to 2 9 .

c 5 2 9 = c 5 × 2 9

3.116
1.

( 7 9 10 5 × 6 3 ) 8

Use the exponent distributive rule. When you have a fraction, a b , raised to an exponent, n, then ( a b ) n = a n b n .

( 7 9 ) 8 ( 10 5 × 6 3 ) 8

Use the distributive rule for exponents in the denominator: ( a × b ) n = a n × b n .

( 7 9 ) 8 ( 10 5 ) 8 × ( 6 3 ) 8

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents.

( a n ) m = a ( n × m )

7 72 10 40 × 6 24

7 72 10 40 × 6 24
2.

( 4 a 9 b 6 ) 2

Use the exponent distributive rule. When you have a fraction, a b , raised to an exponent, n, then ( a b ) n = a n b n .

( 4 ) 2 ( a 9 b 6 ) 2

Use the distributive rule for exponents in the denominator: ( a × b ) n = a n × b n .

( 4 ) 2 ( a 9 ) 2 × ( b 6 ) 2

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents.

( a n ) m = a ( n × m )

16 a 18 b 12

16 a 18 b 12
3.117
1.

This is not a product of a number between 1 and 9 multiplied by 10 raised to an integer power.

42.67 is bigger than 9.

Is not written in scientific notation; 42.67 is not at least 1 and less than 10.
2.
This is a product of a number between 1 and 9 multiplied by 10 raised to an integer power.
Is written in scientific notation
3.

This is not a product of a number between 1 and 9 multiplied by 10 raised to an integer power.

80.91 is bigger than 9.

Is not written in scientific notation; The absolute value of –80.91 is not at least 1 and less than 10.
3.118
1.

–38300

Case 3: The absolute value of the number is 10 or larger.

Step 1: Count the number of digits that are to the left of the decimal point. Label this n.

n = 5

Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, include the sign.

–38300

Step 3: If there is more than one digit, place the decimal point after the first digit.

–3.8300

Step 4: Multiply the number from Step 3 by 10 n 1 .

3.83 × 10 4

3.38 × 10 4
2.

0.0045

Case 2: The absolute value of the number is less than 1.

Step 1: Count the number of digits between the decimal and the first non-zero digit. Label this n.

n = 2

Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the negative sign.

45

Step 3: If there is more than one digit, place the decimal after the first digit from Step 2.

4.5

Step 4: Multiply the number from 3 by 10 ( n + 1 ) .

4.5 × 10 ( 2 + 1 ) = 4.5 × 10 3

4.5 × 10 3
3.

1

Case 1: The number is a single-digit integer.

In this case, the scientific notation form of the number is d i g i t × 10 0 .

1 × 10 0

1 × 10 0
3.119
1.

46.113 × 10 8

When you move the decimal point four places to the left, 46.113 becomes

0.0046113, or 4.6113 × 10 4 .

To keep the number the same, you need to multiply by 104. Increase the exponent by 4 to make up for moving the decimal place four places to the left.

46.113 × 10 8 = 0.0046113 × 10 12

0.0046113 × 10 12
3.120
1.

149.11 × 10 4

When you move the decimal point two places to the right, 149.11 becomes 14,911. Because the number became larger, you need to decrease the exponent by 2. Change the exponent from –4 to –6.

14911.0 × 10 6

14 , 911.0 × 10 6
3.121
1.

Because the exponent is positive, move the decimal point six places to the right. (Think of a number line. The positive end is to the right.)

1,020,000

1,020,000
2.

Because the exponent is negative, move the decimal point six places to the left. (Think of a number line. The negative end is to the left.)

0.0000409

0.0000409
3.122
1.

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, add the other part of the numbers separately.

7.57 × 10 13 + 2.031 × 10 13

( 7.57 + 2.031 ) × 10 13

9.601 × 10 13

9.601 × 10 13
2.

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, subtract the other part of the numbers separately.

3.03 × 10 6 1.5 × 10 6

( 3.03 1.5 ) × 10 6

1.53 × 10 6

1.53 × 10 6
3.123
1.

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, add the other part of the numbers separately.

5.08 × 10 3 + 6.9 × 10 3

( 5.08 + 6.9 ) × 10 3

11.98 × 10 3

1.198 × 10 4

1.198 × 10 4
2.

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, subtract the other part of the numbers separately.

8.968 × 10 38 8.761 × 10 38

( 8.968 8.761 ) × 10 38

0.207 × 10 38

2.07 × 10 39

2.07 × 10 39
3.124
1.

1.14 × 10 43 + 2.56 × 10 46

First, you need the two numbers to have the same power of 10.

Make them both have an exponent of –43. Move the decimal point three places to the left for the second number.

0.00256 × 10 43

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, add the other part of the numbers separately.

1.14 × 10 43 + 0.00256 × 10 43

( 1.14 + 0.00256 ) × 10 43

( 1.14 + 0.00256 ) × 10 43

1.14256 × 10 43

1.14256 × 10 43
3.125
1.

9.15 × 10 28 7.23 × 10 26

First, you need the two numbers to have the same power of 10.

Make them both have an exponent of 28. Move the decimal point two places to the left for the second number.

0.0723 × 10 28

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, subtract the other part of the numbers separately.

( 9.15 0.0723 ) × 10 28

9.0777 × 10 28

9.0777 × 10 28
3.126
1.

( 2.29 × 10 3 ) × ( 3 × 10 4 )

Step 1: Multiply the number parts.

2.29 × 3 = 6.87

Step 2: Add the exponents of the 10s.

3 + 4 = 7

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

6.87 × 10 7

Step 4: If the answer is not in scientific notation, adjust it.

Not necessary.

6.87 × 10 7

6.87 × 10 7
2.

( 6.91 × 10 3 ) × ( 9.1 × 10 5 )

Step 1: Multiply the number parts.

6.91 × 9.1 = 62.881

Step 2: Add the exponents of the 10s.

3 + 5 = 2

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

62.881 × 10 2

Step 4: If the answer is not in scientific notation, adjust it.

6.2881 × 10 3

6.2881 × 10 3
3.127
1.

( 3.6 × 10 2 ) / ( 1.5 × 10 3 )

Step 1: Divide the number parts.

3.6 ÷ 1.5 = 2.4

Step 2: Subtract the exponents of the 10s.

2 3 = 5

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

2.4 × 10 5

Step 4: If the answer is not in scientific notation, adjust it.

2.4 × 10 5

2.4 × 10 5
2.

( 1.8 × 10 4 ) / ( 4.8 × 10 3 )

Step 1: Divide the number parts.

1.8 ÷ 4.8 = 0.375

Step 2: Subtract the exponents of the 10s.

4 3 = 1

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

0.375 × 10 1

Step 4: If the answer is not in scientific notation, adjust it.

3.75 × 10 0

3.75 × 10 0
3.128
1.

Subtract 2 × 10 10 from 1.4 × 10 8 .

First, you need to make the powers of 10 match.

Make them both have an exponent of –8. Move the decimal point two places to the left for the first number.

0.02 × 10 8

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, subtract the other part of the numbers separately.

1.4 × 10 8 0.02 × 10 8

( 1.4 0.02 ) × 10 8

1.38 × 10 8

The transistor is 1.38 × 10 8 meters larger than the diameter of an atom.

The transistor is 1.38 × 10 8 m larger than the diameter of an atom.
3.129
1.

2.781 × 10 9 / 3.114 × 10 7

Step 1: Divide the number parts.

2.781 ÷ 3.114 0.8930635

Step 2: Subtract the exponents of the 10s.

9 7 = 2

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

0.89306 × 10 2

Step 4: If the answer is not in scientific notation, adjust it.

8.9306 × 10 1

Neptune is approximately 8.9306 × 10 1 , or 89.3, times farther from the sun than Mercury.

Neptune is 8.930 × 10 1 , or 89.3, times further from the sun that Mercury.
3.130
1.

Divide 7.5 × 10 21 by 1 × 10 12 .

Step 1: Divide the number parts.

7.5 ÷ 1 = 7.5

Step 2: Subtract the exponents of the 10s.

21 12 = 9

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

7.5 × 10 9

Step 4: If the answer is not in scientific notation, adjust it.

7.5 × 10 9

There are 7.5 × 10 9 cubic meters of sand on the Australian coastline.

7.5 × 10 9 cubic meters
3.131
1.

Divide 6.4235 × 10 12 by 7.647 × 10 9 .

Step 1: Divide the number parts.

6.4235 ÷ 7.647 0.84019877

Step 2: Subtract the exponents of the 10s.

12 9 = 3

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

0.84 × 10 3

Step 4: If the answer is not in scientific notation, adjust it.

8.4 × 10 2

A human exhales approximately 8.4 × 10 2 pounds of carbon dioxide per year.

A person exhales, on average, 8.4 × 10 2 , or 840 pounds of carbon dioxide per year.
3.132
1.

7.6 2.2 = 5.4

5.4 2.2 = 3.2

3.2 2.2 = 1.0

All the terms follow this pattern, so this is an arithmetic sequence.

This is an arithmetic sequence. Every term is the previous term minus 2.2.
2.

14 + 2 = 16

16 + 2 = 18

18 + 2 = 20 , but that is not the next term.

This is not an arithmetic sequence.

This is not an arithmetic sequence. The difference between terms 1 and 2 is 2, but between terms 3 and 4 the difference is 4. The differences are not the same.
3.

14 + 6 = 20

20 + 6 = 26

26 + 6 = 32

This pattern continues for all the terms shown. This is an infinite arithmetic sequence.

This is an (infinite) arithmetic sequence. Every term is the previous term plus 6. The ellipsis indicates the pattern continues.
3.133
1.

The term a 1 is the first term in a sequence. In this sequence, a 1 = 4.5 .

The term, d , represents the constant difference. Subtract the first term from the second term.

8.1 4.5 = 3.6

d = 3.6

You can find any term by using the formula: a i = a 1 + d × ( i 1 ) .

a 86 = 4.5 + 3.6 × ( 86 1 ) = 4.5 + 3.6 ( 85 ) = 310.5

a 1 = 4.5 , d = 3.6 , a 36 = 310.5
3.134
1.

a 14 = 41 , a 38 = 161

When you know two terms, you can find the constant difference using this formula: d = a j a i j i .

d = 161 41 38 14 = 120 24 = 5

The first term in the sequence can be found using this formula: a 1 = a i d ( i 1 ) and a 14 = 41 .

a 1 = 41 5 ( 14 1 ) = 41 5 ( 13 ) = 41 65 = 24

The 151st term can be found using this formula: a i = a 1 + d × ( i 1 ) .

a 151 = 24 + 5 × ( 151 1 ) = 24 + 5 ( 150 ) = 24 + 750 = 726

The constant difference is 5, the first term is 24 , and the 151st term is 726.

d = 5 , a 1 = 24 , and a 151 = 726
3.135
1.

The sum of the first n terms of an arithmetic sequence is: s n = n ( a 1 + a n 2 ) .

You do not know the last term, so first use a i = a 1 + d × ( i 1 ) to find the 101st term.

a 101 = 13 + 2.25 × ( 101 1 ) = 13 + 2.25 ( 100 ) = 13 + 225 = 238

Now, you can use the sum formula.

s n = 101 ( 13 + 238 2 ) = 101 ( 251 2 ) = 12 , 675.5

12,675.5
3.136
1.

First week: $10

Second week: $15

Third week: $20

This is an arithmetic sequence where the first term is 10 and the constant difference is 5.

Use this formula to find the 52nd term: a i = a 1 + d × ( i 1 ) .

a 52 = 10 + 5 × ( 52 1 ) = 10 + 5 ( 51 ) = 10 + 255 = 265

Christina will save $265 in the 52nd week.

Christina will save $265 in week 52.
3.137
1.

This is an arithmetic sequence where the first term is 24 and the constant difference is 2.

You want to know the sum of the first 40 terms.

You first need to know the 40th term to use the sum formula.

You can find the 40th term by using the formula: a i = a 1 + d × ( i 1 ) .

a 40 = 24 + 2 × ( 40 1 ) = 24 + 2 ( 39 ) = 24 + 78 = 102

Find the sum of the first 40 rows: s n = n ( a 1 + a n 2 ) .

s 40 = 40 ( 24 + 102 2 ) = 40 ( 126 2 ) = 2 , 520

The theater has 2,520 seats in the first 40 rows.

There are 2,520 seats in the theater.
3.138
1.

Try 5.

1 × 5 = 5

5 × 5 = 25

25 × 5 = 125

The pattern continues to work, so this is a geometric sequence. The common ratio is 5.

It is a geometric sequence; common ratio is 5.
2.

Try –2.

3 × 2 = 6

6 × 2 = 12

12 × 2 = 24

24 × 2 = 48 , which is not the next term. This is not a geometric sequence.

It is not a common ratio; term 2 is the first term multiplied by −2, but the sixth term is the fifth term multiplied by 3.
3.

Try –0.1.

500 × 0.1 = 50

50 × 0.1 = 5

The pattern continues to work. This is a geometric sequence. The common ratio is –0.1 or 1 10 .

It is a geometric sequence; common ratio is 1 10 .
3.139
1.

The nth term of a geometric sequence, a n , with first term a 1 and common ratio r, is a n = a 1 r n 1 .

a 12 = 3072 ( 1 2 ) 11 = 3 2

Alternate answer option: 1.5

3 2
2.

The nth term of a geometric sequence, a n , with first term a 1 and common ratio r, is a n = a 1 r n 1 .

a 5 = 0.5 ( 8 ) 4 = 2 , 048

2,048
3.140
1.

The sum of the first n terms of a geometric sequence, with first term a 1 and common ratio r, is

s n = a 1 ( 1 r n 1 r ) , provided that r 1 .

s 10 = 7 ( 1 6 10 1 6 ) = 84 , 652 , 645

84,652,645
2.

The sum of the first n terms of a geometric sequence, with first term a 1 and common ratio r, is

s n = a 1 ( 1 r n 1 r ) , provided that r 1 .

s 6 = 27 ( 1 ( 1 3 ) 6 1 1 3 ) = 364 9 40.4444

40.444444
3.141
1.

If you deposit P dollars in an account that earns interest compounded yearly, then the amount in the account, A, after t years is calculated with the formula: A = P ( 1 + r ) t . This is a geometric sequence with constant ratio (1 + r) and first term a 1 = P .

A = P ( 1 + r ) t = 4 , 000 ( 1 + 0.055 ) 20 $ 11 , 671.03

The amount in the account was $11,671.03 (rounded to two decimal places).
3.142
1.

You can use geometric sequences to interpret exponential growth when numbers double every 30 minutes. The common ratio is 2. The first term is 15. The time interval is 30 minutes. In 20 hours, there are 40 of your 30-minute intervals.

a 20 = 15 ( 2 ) 19 1.6493 × 10 13 bacteria

There are 1.6493 × 10 13 organisms after 20 hours.
3.143
1.

First term: 0.5 of square is blue, 0.5 is not blue.

Second term: You’ve colored half of the remaining area, so 0.75 is blue. 0.25 is not blue.

Third term: You’ve colored half of the remaining area. 0.875 is blue; 0.125 is not blue.

Fourth term: You’ve colored half of the remaining area. 0.9375 is blue; 0.0625 is not blue.

The common ratio for what is not blue is 0.5.

The blue numbers are not a geometric sequence, but the not blue numbers are. You can find the 15th term of the not blue numbers and subtract that from 1.

1 a 15 = 1 a 1 r n 1 = 1 0.5 ( 0.5 ) 14 0.9999694824

0. 99996948242188

Check Your Understanding

1.

31 and 701 are prime.

The rest are composite: 56, 213, 48.

56 = 2 × 2 × 2 × 7

213 = 3 × 71

48 = 2 × 2 × 2 × 2 × 3

31 and 701 are prime. 56, 213 and 48 are composite.
2.

4,570 ends in 0, so it has a factor of 10. Divide out 2 and 5.

4 , 570 = 2 × 5 × 457

The square root of 457 is approximately 21.37.

457 is not divisible by 2, 3, 5, 7, 11, 13, 17, or 19, the primes less than 21.

Thus, you are done and 4 , 570 = 2 × 5 × 457 .

2 × 5 × 457
3.
Number Factors    
410 2           5 41
144 2 2 2 2 3 3    
Common 2              

The greatest common divisor is the product of the common factors. There is only one common factor: 2.

2
4.

Find the LCM of 45 and 70.

Find the prime factorization of each number.

45 = 3 2 × 5

70 = 2 × 5 × 7

Make a table of each prime and the largest exponent for each prime.

Prime 2 3 5 7
Exponent 1 2 1 7

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is 2 × 3 2 × 5 × 7 = 630 .

630
5.

Find the GCD of 30, 20, and 70.

Number Factors        
30 2   3 5  
20 2 2 5  
70 2   5 7
Common 2     5  

Greatest common divisor = 2 × 5 = 10

The maximum number of bags is 10.

The maximum number of bags that can be filled in this way is 10.
6.

–4 is an integer because it is the negative of a counting number.

15.2 is not an integer as it has a decimal part. It is between two integers, 15 and 16.

The square root of 2 is not an integer. It is approximately 1.414, which is between two integers, 1 and 2.

The fraction 3 divided by 20 is not an integer. It is equal to 0.15, which is between two integers, 1 and 2.

430 is an integer. It is a counting number.

−4, 430
7.

Use a number line marked with tick marks to represent every integer.

4: Place a solid dot four tick marks to the right of zero.

–2: Place a solid dot two tick marks to the left of zero.

7: Place a solid dot seven tick marks to the right of zero.

A number line ranges from negative 7 to 7, in increments of 1. Three points are marked at negative 2, 4, and 7. The points are labeled x equals negative 2, x equals 4, and x equals 7, respectively.
8.

Consider a number line marked with tick marks to represent every integer. The farther a number is to the left, the less its value.

The negative numbers are on the left of zero.

The number farthest to the left is –13, then –7, then –2.

The positive numbers are to the right of zero.

Of the positive numbers, 4 is left of 10.

In increasing order: –13, –7, –2, 4, 10

−13, −7, −2, 4, 10
9.
7

Because the value inside the absolute value bars is negative, the absolute value removes the negative sign.

| 7 | = 7

10.
Subtracting a negative number is the same as adding a positive number. 4 – (–9) is the same as adding 4 + 9 to get 13.
13
11.
36

( 3 ) × ( 12 ) = 36

Because –3 and –12 have the same sign, the product is positive.

Section 3.2 Exercises

12.

Parentheses have the highest precedence.

Order of Operations (PEMDAS)

Parentheses

Exponents

Multiplication/Division

Addition/Subtraction

For same order operations, work left to right.

parentheses
13.

Exponents are performed before addition.

Order of Operations (PEMDAS)

Parentheses

Exponents

Multiplication/Division

Addition/Subtraction

For same order operations, work left to right.

exponents
14.

2 × 3 2 5 × 8

Use PEMDAS.

Evaluate the exponent. 2 × 9 5 × 8
Multiply by 9. 18 5 × 8
Multiply by 8. 18 40
Subtract. 22
−22
15.
parentheses
16.

( 4 3 ) 2 + 27 × 8 2 ÷ 6 2

Use PEMDAS.

Subtract inside the parentheses. ( 1 ) 2 + 27 × 8 2 ÷ 6 2
Calculate the exponents. 1 + 27 × 64 ÷ 36
Multiply by 64. 1 + 1 , 728 ÷ 36
Divide by 36. 1 + 48
Add. 49
49
17.

41 is a rational number as it is an integer. You can write an integer as a fraction over 1.

4 3 is a rational number as it is a ratio of two integers.

2.75 is a rational number as it equals a ratio of two integers. For instance, you can see it is equal to 275 divided by 100.

0.2 13 ¯ is a rational number because repeating decimals are rational numbers.

The square root of 13 is not a rational number because it approximates to a decimal that does not repeat.

41 , 4 3 , 2.75 , 0.2 13 ¯ are rational; 13 is not.
18.
18 30 = 3 × 6 5 × 6 = 3 5
3 5
19.

3 8 + 5 12

What is the LCM of 8 and 12?

Number Factors
8 2 2 2
12 2 2 3
LCM 2 2 2 3

L C M = 2 × 2 × 2 × 3 = 24

The LCM of 8 and 12 is 24.

Rewrite both fractions with 24 as a denominator.

3 × 3 8 × 3 + 5 × 2 12 × 2

9 24 + 10 24

Add the numerators now that the denominators are the same.

19 24

19 24
20.

Step 1: Because 0.34 has two decimal places, write 10 to the second power: 100.

Step 2: Remove the decimal point from the original number: 34.

Step 3: Write a fraction with the result from Step 2 in the numerator and the result from Step 1 in the denominator.

34 100

Alternate answer in lowest terms: 17 50

34 100
21.

47 ÷ 12 is 3 with a remainder. What is that remainder?

12 × 3 = 36

47 36 = 11

The remainder is 11.

So, 47 36 as a mixed number is 3 11 12 .

3 11 12
22.

Multiply the numerators and denominators, dividing out common factors.

2 × 21 9 × 22

2 × 7 × 3 3 × 3 × 2 × 11

7 33

7 33
23.

2 5 ÷ 3 10 + 1 6

Use PEMDAS. Do the division first by multiplying by the reciprocal.

2 5 × 10 3 + 1 6

Multiply the numerators and denominators, dividing out common factors.

2 5 × 2 × 5 3 + 1 6

4 3 + 1 6

Rewrite the first fraction to have 6 in the denominator.

4 × 2 3 × 2 + 1 6

8 6 + 1 6

Add the numerators.

9 6

Divide out common factors.

3 × 3 2 × 3

3 2

3 2
24.

You can use the density property process:

Step 1: Add the two rational numbers.

7 8 + 20 21 = 7 × 21 8 × 21 + 20 × 8 21 × 8

147 168 + 160 168

307 168

Step 2: Divide that result by 2. Dividing by 2 is the same as multiplying by one-half.

307 168 × 1 2 = 307 336

There are other numbers, but this is the one found using this process.

Using the process from the chapter, 307 336 , and there are other answers.
25.

3 2 7

Multiply the integer part by the denominator. 3 × 7 = 21
Add the product to the numerator. 21 + 2 = 23
Write the sum divided by the denominator. 23 7
23 7
26.

Find one-eighth of $882.

1 8 × 882 = 110.25

Divide 882 by 8.

Lina will save $110.25.

$110.25
27.

What is 38% of 600?

n percent, denoted n % , is the fraction n 100 .

Rewrite “of” as multiplication. 38 100 × 600
Use your calculator. 228
228
28.

What is 10% of 70?

n percent, denoted n % , is the fraction n 100 .

Rewrite “of” as multiplication. 10 100 × 70
Use your calculator. 7

The company will hire 7 new employees.

7 new employees will be hired.
29.

500 = 100 5 = 10 5

10 5
30.

3 7 10 7

Because they have the same irrational part, you can subtract without using a calculator.

The formula uses the distributive property: ( a × x ) ± ( b × x ) = ( a ± b ) × x .

( 3 10 ) × 7 Subtract.

7 7

7 7
31.

8 10 × ( 3 2 )

Step 1: Multiply the rational part.

8 × 3 = 24

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Multiply the irrational parts.

10 × 2 = 20

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

20 = 4 5 = 2 5

Step 5: The result is the product of the rational and the irrational parts.

24 ( 2 5 ) = 48 5

48 5
32.

4 7

Multiply the numerator and denominator by 7 .

4 × 7 7 × 7

Simplify.

4 7 7

Section 3.5 Exercises

4 7 7
33.

Real numbers: 77 , 19 , 38.902

Not real: 4 + i is not real because any number that contains i is not a real number.

77 , −19 , 38.902
34.
N Z N Q N R Z Q Z R Q R
35.

When using the distributive property, multiplication distributes across addition.

a × ( b + c ) = ( a × b ) + ( a × c )

The rule can be written in the other direction.

( a × b ) + ( a × c ) = a × ( b + c )

distributive property
36.

7 + 19 = 26

Find 23 mod 12.

Determine the remainder when 26 is divided by 12.

26 ÷ 12 = 2 and a remainder. What is that remainder?

12 × 2 = 24

Subtract 24 from 26.

26 24 = 2

The remainder is 2.

The remainder is the solution to 26 mod 12.

26 mod 12 ≡ 2

2
37.

8 31 = 23

Find –23 mod 12.

It is easier to think about modulo problems when the number is positive. The hand is in the same position every 12 hours, so you can always add any multiple of 12 hours. Add 24 hours to make the number positive.

23 + 24 = 1

Now, find 1 mod 12.

Determine the remainder when 1 is divided by 12.

1 ÷ 12 = 0 and a remainder. What is that remainder?

The remainder is 1.

The remainder is the solution to 1 mod 12.

1 mod 12 ≡ 1

–23 mod 12 = 1

The answer to 8 – 31 is 1.

1
38.

5 × 37 = 185

Find 185 mod 12.

Determine the remainder when 185 is divided by 12.

185 ÷ 12 = 15 and a remainder. What is that remainder?

12 × 15 = 180

Subtract 180 from 185.

185 180 = 5

The remainder is 5.

The remainder is the solution to 185 mod 12.

185 mod 12 ≡ 5

5
39.

If Calene calls every fourth day and she calls eight times, then 32 days have passed.

Find 32 mod 7.

To find what day it is, you need to find 32 modulo 7.

32 ÷ 7 = 4 and a remainder. What is that remainder?

7 × 4 = 28

Subtract 28 from 32.

32 28 = 4

The remainder is 4.

The remainder is the solution to 32 modulo 7.

32 modulo 7 ≡ 4

Four days after Monday is Friday.

It will be Friday when Calene calls her mother again.

Friday
40.

Use the product rule: a n × a m = a n + m .

a 3 × a 5 = a 3 + 5 = a 8

a 8
41.

Use the quotient rule: a n a m = a ( n m ) .

5 4 5 8 = 5 ( 4 8 ) = 5 4

Alternate answer option:

Use the negative exponent rule. a n = 1 a n provided that a ≠ 0.

5 4 = 1 5 4

5 4 or 1 5 4
42.

( 6 b ) 9

Use the distributive rule for exponents: ( a × b ) n = a n × b n .

( 6 b ) 9 = 6 9 b 9

6 9 b 9
43.

( c 7 ) 3

Use the exponent distributive rule. When you have a fraction, a b , raised to an exponent, n, then ( a b ) n = a n b n .

( c 7 ) 3 = c 3 7 3

c 3 7 3
44.

( 3 a 2 4 b 5 ) 6

Use the exponent distributive rule. When you have a fraction, a b , raised to an exponent, n, then ( a b ) n = a n b n .

( 3 a 2 4 b 5 ) 6 = ( 3 a 2 ) 6 ( 4 b 5 ) 6

Use the distributive rule for exponents: ( a × b ) n = a n × b n .

( 3 a 2 ) 6 ( 4 b 5 ) 6 = 3 6 ( a 2 ) 6 4 6 ( b 5 ) 6

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents.

( a n ) m = a ( n × m )

3 6 ( a 2 ) 6 4 6 ( b 5 ) 6 = 3 6 a 12 4 6 b 30

3 6 a 12 4 6 b 30
45.

0.00456

Case 2: The absolute value of the number is less than 1.

Step 1: Count the number of digits between the decimal and the first non-zero digit. Label this n.

n = 2

Step 2: Starting with the first non-zero digit of the number, write the digits. If the number was negative, include the negative sign.

456

Step 3: If there is more than one digit, place the decimal after the first digit from Step 2.

4.56

Step 4: Multiply the number from 3 by 10 ( n + 1 ) .

4.56 × 10 3

4.56 × 10 3
46.

Because the exponent is positive, move the decimal point eight places to the right. (Think of a number line. The positive end is to the right.)

567,000,000

567 , 000 , 000
47.

4.5 × 10 3 + 9.8 × 10 2

First, you need to make the powers of 10 match.

Make them both have an exponent of 3. Move the decimal point one place to the left on the second number.

0.98 × 10 3

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, add the other part of the numbers separately.

4.5 × 10 3 + 0.98 × 10 3

( 4.5 + 0.98 ) × 10 3

5.48 × 10 3

5.48 × 10 3
48.

2.5 × 10 5 9.8 × 10 6

First, you need to make the powers of 10 match.

Make them both have an exponent of 6. Move the decimal point one place to the left on the first number.

0.25 × 10 6

Because the powers of 10 match, use the distributive property to factor the power of 10 from the numbers. Then, subtract the other part of the numbers separately.

0.25 × 10 6 9.8 × 10 6

( 0.25 9.8 ) × 10 6

9.55 × 10 6

9.55 × 10 6
49.

( 7.4 × 10 4 ) × ( 4.8 × 10 3 )

Step 1: Multiply the number parts.

7.4 × 4.8 = 35.52

Step 2: Add the exponents of the 10s.

4 + 3 = 7

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

35.52 × 10 7

Step 4: If the answer is not in scientific notation, adjust it.

3.552 × 10 8

3.552 × 10 8
50.
5.75 × 10 3

4.6 × 10 4 8 × 10 8

Step 1: Divide the number parts.

4.6 ÷ 8 = 0.575

Step 2: Subtract the exponents of the 10s.

4 ( 8 ) = 4

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

0.575 × 10 4

Step 4: If the answer is not in scientific notation, adjust it.

5.75 × 10 3

51.

Divide to find the answer: 1.514 × 10 10 ÷ 4.3 × 10 3 .

Step 1: Divide the number parts.

1.514 ÷ 4.3 0.3520930233

Step 2: Subtract the exponents of the 10s.

10 ( 3 ) = 13

Step 3: The result is the answer from Step 1 times 10 raised to the answer from Step 2.

0.3521 × 10 13

Step 4: If the answer is not in scientific notation, adjust it.

3.521 × 10 12

A pile of dollar bills that reaches the moon would contain approximately 3.521 × 10 12 dollar bills.

A pile of dollar bills that reaches the moon would contain 3.521 × 10 12 bills.
52.

3 + 3 = 6

6 + 3 = 9

9 + 3 = 12 , but the next term is not 12. It’s 15. This is not an arithmetic sequence.

No. The difference from term 1 to term 2 is different than the difference from term 4 to term 5.
53.

Count to the seventh term in the sequence.

1st term: 1

2nd term: 5

3rd term: 7

4th term: 100

5th term: 4

6th term: –17

7th term: 8

8
54.

You can find any term by using the formula: a i = a 1 + d × ( i 1 ) .

a 135 = 10 + 4.5 × ( 135 1 ) = 10 + 4.5 ( 134 ) = 10 + 603 = 613

613
55.

a 8 = 35 , a 40 = 131

When you know two terms, you can find the constant difference using this formula: d = a j a i j i .

d = 131 35 40 8 = 96 32 = 3

The first term in the sequence can be found using this formula: a 1 = a i d ( i 1 ) and a 8 = 35 .

a 1 = 35 3 ( 8 1 ) = 35 3 ( 7 ) = 35 21 = 14

The constant difference is 3 and the first term is 14.

d = 3 , a 1 = 14
56.

You need to know the 100th term.

You can find any term by using the formula: a i = a 1 + d × ( i 1 ) .

a 100 = 4 + 7 × ( 100 1 ) = 4 + 7 ( 99 ) = 697

The sum of the first n terms of an arithmetic sequence is: s n = n ( a 1 + a n 2 ) .

s 100 = 100 ( 4 + 697 2 ) = 100 ( 701 2 ) = 35 , 050

35050
57.

This is an arithmetic sequence with a first term of 30 and a constant difference of 4. The answer to the question is the 100th term.

You can find any term by using the formula: a i = a 1 + d × ( i 1 ) .

a 100 = 30 + 4 × ( 100 1 ) = 30 + 4 ( 99 ) = 426

There will be 426 people in the group on the 100th day.

There will be 426 people in their survey group after 100 days.
58.

Try 2.

3 × 2 = 6

6 × 2 = 12

12 × 2 = 24

The pattern continues for each term. Each term is 2 times the previous term. This is a geometric sequence.

Yes, each term is the previous term multiplied by 2.
59.

To find the common ratio, divide a term by the previous term.

30 3 = 10

The common ratio is −10.
60.

The nth term of a geometric sequence, a n , with first term a 1 and common ratio r, is a n = a 1 r n 1 .

a 15 = 10 ( 1.5 ) 14 2 , 919.293 (rounded to three decimal places)

2,919.293 (rounded off to three decimal places)
61.

The sum of the first n terms of a geometric sequence, with first term a 1 and common ratio r, is

s n = a 1 ( 1 r n 1 r ) , provided that r 1 .

s 100 = 4 ( 1 0.3 100 1 0.3 ) 5.714 (rounded to three decimal places)

5.714 (rounded to three decimal places)
62.

If you deposit P dollars in an account that earns interest compounded yearly, then the amount in the account, A, after t years is calculated with the formula: A = P ( 1 + r ) t . This is a geometric sequence with constant ratio (1 + r) and first term a 1 = P .

A = P ( 1 + r ) t = 15 , 000 ( 1 + 0.042 ) 17 $ 30 , 188.57 (rounded to two decimal places)

$30,188.57 (rounded off to two decimal places)
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