How to find fractional percentages - ACT Math

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Question

Convert the following to a percentage.

$\frac{18}{4}$

Answer

To convert a fraction into a percentage, simply simplify the fraction into a decimal and multiply by 100.

$\frac{18}{4}\rightarrow4\frac{2}{4}\rightarrow 4.5$

$4.5\cdot100=450%$

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Question

The girl's tennis team decided to do multiple fundraisers to raise enough to buy new equipment for next season. The following is a table of 4 of the fundraisers they did and the fraction of the total funds each raised.

The last fundraiser they did earned them the final $\$257$ they needed. What percent of the total funds raised did the last fundraiser raise?

Answer

The question asks to find what percent the last fundraiser makes up of the total funds raised. The information that's given is how much that last fundraiser made and what fraction the other fundraisers made.

This problem may be solved without the amount of money the last fundraiser raised. The reason is that we know that only 5 fundraisers were done. We're given what the fraction of the other fundraisers were.

If we were to add all the fractions together, including the last fundraiser, the sum of all would equal 1.

$\frac{2}{7}+\frac{1}{8}+\frac{2}{7}+\frac{1}{16}+x=1$

Fractions are another version of decimals but also another version of percentages. Percentages are essentially a prettier way of showing parts (or fractions) of a whole, which would be 100%. In the case of fractions, this is the same logic, which is why when all the fractions are summed, they equal 1. This 1 represents 100%.
Upon using lowest common denominators, adding the fractions together, and solving for x, x is calculated to be $x=\frac{27}{112}$

We've almost obtained the final answer. Now it's just a matter of converting this fraction into a percent. As mentioned previously, fractions are another version of decimals. Think of decimals as being the middle conversion between fractions and percents. All that needs to be done to a decimal to turn it in to a percent is to multiply it by 100. For instance, .50 becomes 50%. Conversely, .50 is the same as $\frac{1}{2}$ in fraction form. Therefore, one half is 50%.
In order to solve for x in percent, the same conversion must be done.
$x=\frac{27}{112}=0.241$
$0.241\cdot 100= 24.1% = x$

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Question

If a dress in a store originally costs $\small \$80$ , what does it cost (in dollars) after a $\small 35%$ discount is applied?

Answer

To find $\small 35%$ of 80, $\small 80\times 0.35=28$ . Next, subtract $\small 80-28=52$ , for $\small \$52$ as the final answer.

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Question

There are students in a math class. $\frac{3}{5}$ of the class consists of boys. How many girls are in the math class?

Answer

First, we need to find the fraction of the class that are girls.

$\frac{5}{5}-\frac{3}{5}=\frac{2}{5}$

Next, find the number of students that are female. If we know that there are 300 students total, we can write the following ratio for girls:

$\frac{x}{300}$

Create a proportion.

$\frac{x}{300}=\frac{2}{5}$

Cross multiply and solve for the number of girls, .

$\frac{x}{300}\times\frac{2}{5}$

$5\cdot x=2\cdot300$

Divide both sides of the equation by 5.

$\frac{5x}{5}=\frac{600}{5}$

There are 120 girls in the class.

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Question

On a acre plot of land, there are acres in which deer live. What percentage of the land does this represent? Round to the nearest hundredth.

Answer

For percentages, it is always easiest to translate into language that uses "is" and "of." "Is" means "equals" and "of" means "multiply." So, we have:

What percentage of is ? This is the same as:

Solving for , you get:

$x=\frac{7}{15000}=0.00046666666667$

Now, as a percent, this is equal to:

Rounding to the nearest hundredth, you have .

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Question

is $8\frac{1}{3} %$ of , which is 35% of . All three are positive integers. In which range does the smallest possible value of fall?

Answer

is 35% of , so $Y = \frac{35}{100}Z = \frac{7}{20}Z$ .

is $8\frac{1}{3} %$ of , so $X = \frac{8\frac{1}{3}}{100}Y = \frac{8\frac{1}{3}\times 3}{100 \times 3}Y = \frac{25}{300 }Y = \frac{1}{12}Y$ .

Consequently,

$X = \frac{1}{12}Y = \frac{1}{12} \cdot \frac{7}{20}Z = \frac{7}{240}Z$

The smallest integer can be is 240. If this happens,

$Y = \frac{7}{20}Z = \frac{7}{20} \cdot 240 = 84$

$X = \frac{7}{240} \cdot 240 = 7$

Their sum is

The correct choice is that the sum is between 300 and 400.

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Question

is $16\frac{2}{3} %$ of and $62\frac{1}{2} %$ of ; all are positive integers. Give the smallest possible value of .

Answer

is $16\frac{2}{3} %$ of and $62\frac{1}{2} %$ of , so

$A = \frac{16\frac{2}{3} }{100}B = \frac{16\frac{2}{3}\cdot 3 }{100 \cdot 3}B = \frac{50}{300}B = \frac{1}{6} B$ ,

so .

$A = \frac{62\frac{1}{2} }{100}C = \frac{62\frac{1}{2} \cdot 2 }{100 \cdot 2}C= \frac{125}{200}C = \frac{5}{8} C$

so $C = \frac{8} {5}A$

must be divisible by 5, so the least possible value is 5 itself.

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Question

is $7\frac{1}{2} %$ of . Which of the following expressions is equal to ?

Answer

is $7\frac{1}{2} %$ of , so

$M = \frac{7\frac{1}{2}}{100} \cdot N = \frac{7\frac{1}{2} \cdot 2}{100\cdot 2} \cdot N = \frac{15}{200} N = \frac{3}{40} N$

$N = \frac{40}{3}M = \frac{40M}{3}$

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Question

.

is $15 \frac{5}{8} %$ of , which is 80% of . What percent of is ?

Answer

is 80% of , so $N = \frac{80 }{100}P = \frac{4}{5}P$ .

Consequently, $P = \frac{5}{4}N$

is $15 \frac{5}{8} %$ of , so $M = \frac{15 \frac{5}{8} }{100} N = \frac{15 \frac{5}{8} \cdot 8 }{100\cdot 8} N = \frac{5 }{32} N$

Consequently, $N = \frac{32} {5 }M$ .

$P = \frac{5}{4} \cdot\frac{32} {5 }M = \frac{32}{4}M = 8M$ , so

is 800% of . This is not one of the choices.

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Question

is $22\frac{1}{2} %$ of .

$\frac{2}{M}$ is what percent of $\frac{5}{N}$ ?

Answer

is $22\frac{1}{2} %$ of , so

$M = \frac{22\frac{1}{2}}{100} N= \frac{22\frac{1}{2}\cdot 2}{100\cdot 2} N= \frac{45}{200 } N= \frac{9}{40} N=\frac{9N}{40}$ ,

so

$\frac{1}{M} = \frac{40}{9N}$

and

$\frac{2}{M} = \frac{80}{9N}$

The question can be rewritten as

$\frac{80}{9N}$ is what percent of $\frac{5}{N}$ ?

The answer is

$\frac{\frac{80}{9N}}{\frac{5}{N}} \times 100 %$

$=\left ( \frac{80}{9N} \div \frac{5}{N}} \right ) \times 100 %$

$=\left ( \frac{80}{9N} \cdot \frac{N}{5} \right ) \times 100 %$

$= \frac{16}{9 } \times 100 %$

$= 177 \frac{7}{9} %$

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Question

is $28 \frac{4}{7} %$ of . What percent of is ?

Answer

is $28 \frac{4}{7} %$ of , so

$6M = \frac{28\frac{4}{7}}{100} \cdot 4N = \frac{28\frac{4}{7} \cdot 7 \cdot 4}{100 \cdot 7} \cdot N = \frac{800}{700 } N = \frac{8}{7}N$

$\frac{8}{7}N = 6M$

$\frac{7}{8} \cdot \frac{8}{7}N =\frac{7}{8} \cdot 6M$

$N =\frac{21}{4} M = 5.25 M$ , or 525% of .

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Question

There are registered students in the math club. Students are deciding what activity they could do for their end of semester event. They have three choices: rock climbing, bowling, or outdoor laser tag. students vote for rock climbing, students vote for bowling, and students vote for outdoor laser tag.

What percent of students did not vote?

Answer

The number of students who did not vote is:

The percent of students who did not vote is therefore:

$\frac{6}{48}=\frac{1}{8}=0.125$

of the students did not vote.

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Question

Convert the following to a percentage.

$\frac{18}{4}$

Answer

To convert a fraction into a percentage, simply simplify the fraction into a decimal and multiply by 100.

$\frac{18}{4}\rightarrow4\frac{2}{4}\rightarrow 4.5$

$4.5\cdot100=450%$

Compare your answer with the correct one above

Question

The girl's tennis team decided to do multiple fundraisers to raise enough to buy new equipment for next season. The following is a table of 4 of the fundraisers they did and the fraction of the total funds each raised.

The last fundraiser they did earned them the final $\$257$ they needed. What percent of the total funds raised did the last fundraiser raise?

Answer

The question asks to find what percent the last fundraiser makes up of the total funds raised. The information that's given is how much that last fundraiser made and what fraction the other fundraisers made.

This problem may be solved without the amount of money the last fundraiser raised. The reason is that we know that only 5 fundraisers were done. We're given what the fraction of the other fundraisers were.

If we were to add all the fractions together, including the last fundraiser, the sum of all would equal 1.

$\frac{2}{7}+\frac{1}{8}+\frac{2}{7}+\frac{1}{16}+x=1$

Fractions are another version of decimals but also another version of percentages. Percentages are essentially a prettier way of showing parts (or fractions) of a whole, which would be 100%. In the case of fractions, this is the same logic, which is why when all the fractions are summed, they equal 1. This 1 represents 100%.
Upon using lowest common denominators, adding the fractions together, and solving for x, x is calculated to be $x=\frac{27}{112}$

We've almost obtained the final answer. Now it's just a matter of converting this fraction into a percent. As mentioned previously, fractions are another version of decimals. Think of decimals as being the middle conversion between fractions and percents. All that needs to be done to a decimal to turn it in to a percent is to multiply it by 100. For instance, .50 becomes 50%. Conversely, .50 is the same as $\frac{1}{2}$ in fraction form. Therefore, one half is 50%.
In order to solve for x in percent, the same conversion must be done.
$x=\frac{27}{112}=0.241$
$0.241\cdot 100= 24.1% = x$

Compare your answer with the correct one above

Question

If a dress in a store originally costs $\small \$80$ , what does it cost (in dollars) after a $\small 35%$ discount is applied?

Answer

To find $\small 35%$ of 80, $\small 80\times 0.35=28$ . Next, subtract $\small 80-28=52$ , for $\small \$52$ as the final answer.

Compare your answer with the correct one above

Question

There are students in a math class. $\frac{3}{5}$ of the class consists of boys. How many girls are in the math class?

Answer

First, we need to find the fraction of the class that are girls.

$\frac{5}{5}-\frac{3}{5}=\frac{2}{5}$

Next, find the number of students that are female. If we know that there are 300 students total, we can write the following ratio for girls:

$\frac{x}{300}$

Create a proportion.

$\frac{x}{300}=\frac{2}{5}$

Cross multiply and solve for the number of girls, .

$\frac{x}{300}\times\frac{2}{5}$

$5\cdot x=2\cdot300$

Divide both sides of the equation by 5.

$\frac{5x}{5}=\frac{600}{5}$

There are 120 girls in the class.

Compare your answer with the correct one above

Question

On a acre plot of land, there are acres in which deer live. What percentage of the land does this represent? Round to the nearest hundredth.

Answer

For percentages, it is always easiest to translate into language that uses "is" and "of." "Is" means "equals" and "of" means "multiply." So, we have:

What percentage of is ? This is the same as:

Solving for , you get:

$x=\frac{7}{15000}=0.00046666666667$

Now, as a percent, this is equal to:

Rounding to the nearest hundredth, you have .

Compare your answer with the correct one above

Question

is $8\frac{1}{3} %$ of , which is 35% of . All three are positive integers. In which range does the smallest possible value of fall?

Answer

is 35% of , so $Y = \frac{35}{100}Z = \frac{7}{20}Z$ .

is $8\frac{1}{3} %$ of , so $X = \frac{8\frac{1}{3}}{100}Y = \frac{8\frac{1}{3}\times 3}{100 \times 3}Y = \frac{25}{300 }Y = \frac{1}{12}Y$ .

Consequently,

$X = \frac{1}{12}Y = \frac{1}{12} \cdot \frac{7}{20}Z = \frac{7}{240}Z$

The smallest integer can be is 240. If this happens,

$Y = \frac{7}{20}Z = \frac{7}{20} \cdot 240 = 84$

$X = \frac{7}{240} \cdot 240 = 7$

Their sum is

The correct choice is that the sum is between 300 and 400.

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Question

is $16\frac{2}{3} %$ of and $62\frac{1}{2} %$ of ; all are positive integers. Give the smallest possible value of .

Answer

is $16\frac{2}{3} %$ of and $62\frac{1}{2} %$ of , so

$A = \frac{16\frac{2}{3} }{100}B = \frac{16\frac{2}{3}\cdot 3 }{100 \cdot 3}B = \frac{50}{300}B = \frac{1}{6} B$ ,

so .

$A = \frac{62\frac{1}{2} }{100}C = \frac{62\frac{1}{2} \cdot 2 }{100 \cdot 2}C= \frac{125}{200}C = \frac{5}{8} C$

so $C = \frac{8} {5}A$

must be divisible by 5, so the least possible value is 5 itself.

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Question

is $7\frac{1}{2} %$ of . Which of the following expressions is equal to ?

Answer

is $7\frac{1}{2} %$ of , so

$M = \frac{7\frac{1}{2}}{100} \cdot N = \frac{7\frac{1}{2} \cdot 2}{100\cdot 2} \cdot N = \frac{15}{200} N = \frac{3}{40} N$

$N = \frac{40}{3}M = \frac{40M}{3}$

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