6.1 Understanding Percent

1. Using the definition and $n$ = 18, 18% in fractional form is $\frac{18}{100}$.
2. Using the definition and $n$ = 84, 84% in fractional form is $\frac{84}{100}$.
3. Using the definition and $n$ = 38.7, 38.7% in fractional form is $\frac{38.7}{100}$.
4. Using the definition and $n$ = 213, 213% in fractional form is $\frac{213}{100}$.

1. To convert 17% to its decimal form, divide 17 by 100. This moves the decimal two places to the left, resulting in 0.17. The decimal form of 17% is 0.17.
2. To convert 7% to its decimal form, divide 7 by 100. This moves the decimal two places to the left, resulting in 0.07. The decimal form of 7% is 0.07.
3. To convert 18.45% to its decimal form, divide 18.45 by 100. This moves the decimal two places to the left, resulting in 0.1845. The decimal form of 18.45% is 0.1845.

1. Using the formula and $x$ = 0.34, we calculate $(0.34\times 100)\%$, which gives us 34%.
2. Using the formula and $x$ = 4.15, we calculate $(4.15\times 100)\%$, which gives us 415%.
3. Using the formula and $x$ = 0.0391, we calculate $(0.0391\times 100)\%$, which gives us 3.91%.

1. The total is $x$ = 3,500, and the percent is $n$ = 70. The decimal form of 70% is 0.70. To find the part, or percent of the total, substitute those values into the formula and calculate.
   $$\begin{array}{ccc}\hfill \text{part}& \hfill =\hfill & \mathrm{percent}\times \mathrm{total}\hfill \\ \hfill & \hfill =\hfill & 0.70\times 3500\hfill \\ \hfill & \hfill =\hfill & 2450\hfill \end{array}$$
   From this, we say that 70% of 3,500 is 2,450.
2. The total is $x$ = 720, and the percent is $n$ = 156. The decimal form of 156% is 1.56. To find the part, or percent of the total, substitute those values into the formula and calculate.
   $$\begin{array}{ccc}\hfill \text{part}& \hfill =\hfill & \text{percent}\times \text{total}\hfill \\ \hfill & \hfill =\hfill & 1.56\times 720\hfill \\ \hfill & \hfill =\hfill & \mathrm{1,123.2}\hfill \end{array}$$
   From this, we say that 156% of 720 is 1,123.2.

1. Step 1: The percent is 35, which in decimal form is 0.35. We were given that 35% of the total is 70, so the part is 70. We are to find the total. Substituting into the formula, we have
   $$\begin{array}{ccc}\hfill \text{part}& \hfill =\hfill & \text{percent}\times \text{total}\hfill \\ \hfill 70& \hfill =\hfill & 0.35\times \text{total}\hfill \end{array}$$
   
   Step 2: To find the total, we solve the equation for the total.
   $$\begin{array}{ccc}\hfill 70& \hfill =\hfill & 0.35\times \text{total}\hfill \\ \hfill \frac{70}{0.35}& \hfill =\hfill & \frac{0.35\times \text{total}}{0.35}\hfill \\ \hfill 200& \hfill =\hfill & \frac{\cancel{0.35}\times \text{total}}{\cancel{0.35}}\hfill \\ \hfill 200& \hfill =\hfill & \text{total}\hfill \end{array}$$
   
   From this we see that 200 is the total, or, that 35% of 200 is 70.
   
2. Step 1: The percent is 10, which in decimal form is 0.1. We were given that 10% of the total is 4,000, so the part is 4,000. Substituting into the formula, we have
   $$\begin{array}{ccc}\hfill \text{part}& \hfill =\hfill & \text{percent}\times \text{total}\hfill \\ \hfill \mathrm{4,000}& \hfill =\hfill & 0.1\times \text{total}\hfill \end{array}$$
   
   Step 2: To find the total, we solve the equation for the total.
   $$\begin{array}{ccc}\hfill \mathrm{4,000}& \hfill =\hfill & 0.1\times \text{total}\hfill \\ \hfill \frac{\mathrm{4,000}}{0.1}& \hfill =\hfill & \frac{0.1\times \text{total}}{0.1}\hfill \\ \hfill \mathrm{40,000}& \hfill =\hfill & \frac{\cancel{0.1}\times \text{total}}{\cancel{0.1}}\hfill \\ \hfill \mathrm{40,000}& \hfill =\hfill & \text{total}\hfill \end{array}$$
   
   From this we see that 40,000 is the total, or that 10% of 40,000 is 4,000.

1. Step 1: The total is 500, the percent of the total is 175. Substituting into the formula, we have
   $$\begin{array}{ccc}\hfill \text{part}& \hfill =\hfill & \text{percent}\times \text{total}\hfill \\ \hfill 175& \hfill =\hfill & \text{percent}\times 500\hfill \end{array}$$
   
   Step 2: To find the percent, we solve the equation for the percent.
   $$\begin{array}{ccc}\hfill 175& \hfill =\hfill & \text{percent}\times 500\hfill \\ \hfill \frac{175}{500}& \hfill =\hfill & \frac{\text{percent}\times 500}{500}\hfill \\ \hfill 0.35& \hfill =\hfill & \frac{\text{percent}\times \cancel{500}}{\cancel{500}}\hfill \\ \hfill 0.35& \hfill =\hfill & \text{percent}\hfill \end{array}$$
   
   We see the percent in decimal form is 0.35. Converting from the decimal form yields 35%. We say that 175 is 35% of 500.
   
2. Step 1: The total is 228, the percent of the total is 155. Substituting into the formula, we have
   $$\begin{array}{ccc}\hfill \text{part}& \hfill =\hfill & \text{percent}\times \text{total}\hfill \\ \hfill 155& \hfill =\hfill & \text{percent}\times 228\hfill \end{array}$$
   
   Step 2: To find the percent, we solve the equation for the percent.
   $$\begin{array}{ccc}\hfill 155& \hfill =\hfill & \text{percent}\times 228\hfill \\ \hfill \frac{155}{228}& \hfill =\hfill & \frac{\text{percent}\times 228}{228}\hfill \\ \hfill 0.6798& \hfill =\hfill & \frac{\text{percent}\times \cancel{228}}{\cancel{228}}\hfill \\ \hfill 0.6798& \hfill =\hfill & \text{percent}\hfill \end{array}$$
   
   We see the percent is 0.6798 (rounded to four decimal places). Converting from the decimal form yields 67.98%. We say that 155 is 67.98% of 228.

The percent of students who will return for the 2022 academic year (the retention rate) is 84%. The total number of students who enrolled in the 2021 academic year was 1,350. This means the percent is known and the total is known. From this, we can determine the number of students who will return (percent of the total) for the 2022 academic year using the formula $$\text{part}=\text{percent}\times \text{total}$$. Substituting into the formula and calculating, we find that the number of students that are returning is

So 1,134 students will return for the 2022 academic year.

In this situation, the percent is to be determined. We know the total number of students, 45, and the part of the students that are chemistry majors, 18. Using that information and the formula $$\text{part}=\text{percent}\times \text{total}$$, the percent can be found. Substituting and solving, we have

Converting the 0.4 from decimal form, we find that 40% of the students in the calculus class are chemistry majors.

In this problem, Mariel’s total sales is to be determined. We know the percent she earns is 20%. We also know that her sales commission was $153.00, which is the percent of the total. Using this information and the formula $$\text{part}=\text{percent}\times \text{total}$$ we can find Mariel’s total sales. The decimal form of 20% is 0.2. The part, or percent of the total, is 153. Substituting and solving, we obtain

Mariel’s total sales were $765.00.