ISEE Lower Level Math : How to find a proportion

Study concepts, example questions & explanations for ISEE Lower Level Math

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Example Questions

Example Question #531 : Numbers And Operations

There are 24 students in Mrs. Brown's classroom. 6 of them have red backpacks. What percentage of the students have red backpacks?

Possible Answers:

\(\displaystyle 25\%\)

\(\displaystyle 6\%\)

\(\displaystyle 75\%\)

\(\displaystyle 30\%\)

\(\displaystyle 16\%\)

Correct answer:

\(\displaystyle 25\%\)

Explanation:

Let's first determine the fraction of students in the class that have red backpacks. We can then convert the fraction to a percentage.

The question tells us that 6 out of 24 students have red backpacks. This is equal to six over twenty-four:

\(\displaystyle \frac{6}{24}\)

We can reduce the fraction by removing a common factor.

\(\displaystyle \frac{6}{24}=\frac{6\div6}{24\div6}=\frac{1}{4}\)

The percentage that is equivalent to \(\displaystyle \frac{1}{4}\) is 25 percent, or \(\displaystyle 25\%\).

Therefore, \(\displaystyle 25\%\) is the correct answer. 

Example Question #45 : Ratio And Proportion

Megan buys cookie packs that come in either small or large sizes. The small size has 3 vanilla cookies and 2 chocolate cookies. The large size has the same proportion of vanilla and chocolate cookies. The large comes with 6 chocolate cookies. How many vanilla cookies are in the large box?

Possible Answers:

\(\displaystyle 12\)

\(\displaystyle 4\)

\(\displaystyle 9\)

\(\displaystyle 6\)

\(\displaystyle 7\)

Correct answer:

\(\displaystyle 9\)

Explanation:

We know that there are the same proportions of cookies in both the small and large boxes.

Let's start by looking at the small box. We know that there are 3 vanilla and 2 chocolate cookies, or a ratio of \(\displaystyle 3:2\).

In the large box, we know there are 6 chocolate cookies. If \(\displaystyle x\) is the number of vanilla cookies, then the ratio in the large box is \(\displaystyle x:6\).

The ratios must be equal.

\(\displaystyle \frac{3}{2}=\frac{x}{6}\)

Cross-multiply and solve for \(\displaystyle x\).

\(\displaystyle 3\times6=x\times2\)

\(\displaystyle 18=2x\)

\(\displaystyle 9=x\)

Example Question #11 : How To Find A Proportion

The ratio of men to women in a room is \(\displaystyle 3:4\). If the room has 21 people, and 2 men then leave the room, what is the proportion of men in the room?

Possible Answers:

\(\displaystyle \frac{7}{19}\)

\(\displaystyle \frac{9}{21}\)

\(\displaystyle \frac{7}{21}\)

\(\displaystyle \frac{9}{11}\)

Correct answer:

\(\displaystyle \frac{7}{19}\)

Explanation:

If there is a room of 21 people in which the ratio of men to women is \(\displaystyle 3:4\), that means that there are 9 men and 12 women in the room. 

If 2 men leave the room, there will be 7 men and 12 women in the room, with a total of 19 people. 

Therefore, the proportion of the men in the room is \(\displaystyle \frac{7}{19}\)

Example Question #12 : How To Find A Proportion

If the ratio of boys to girls in a class is \(\displaystyle 1:2\), what fraction of the class is boys?

Possible Answers:

\(\displaystyle \frac{1}{3}\)

\(\displaystyle \frac{1}{2}\)

\(\displaystyle \frac{2}{3}\)

\(\displaystyle \frac{2}{1}\)

\(\displaystyle \frac{1}{4}\)

Correct answer:

\(\displaystyle \frac{1}{3}\)

Explanation:

If the ratio of boys to girls in a class is \(\displaystyle 1:2\), that means that for every 3 students, 1 is a boy.

Looking at the ratio, we see that there is one boy for every two girls. In a group of three students, you would have one boy and two girls.

We can write the fraction of boys as the number of boys over the number of students in the group.

\(\displaystyle \frac{\text{boys}}{\text{total}}=\frac{1}{3}\)

Example Question #12 : How To Find A Proportion

If it takes Linda 6 minutes to complete a multiple choice question, then what percentage of a 10 question multiple choice test will she complete in half an hour?

Possible Answers:

\(\displaystyle 70\)

\(\displaystyle 40\)

\(\displaystyle 50\)

\(\displaystyle 100\)

Correct answer:

\(\displaystyle 50\)

Explanation:

If it takes Linda 6 minutes to complete a multiple choice question, then she will complete \(\displaystyle 30\div6=5\) questions in 30 minutes. 

Therefore, Linda will complete 5 out of 10 questions in half an hour, or 50%. 

Example Question #11 : How To Find A Proportion

On a typical day, twenty percent of the pies that a pie store sells are blueberry pies. If the store sells forty pies on just such a typical day, how many blueberry pies were likely sold?

Possible Answers:

\(\displaystyle 6\)

\(\displaystyle 10\)

\(\displaystyle 8\)

\(\displaystyle 4\)

Correct answer:

\(\displaystyle 8\)

Explanation:

If  store sells \(\displaystyle 40\) pies, and typically \(\displaystyle 20\%\) of the pies that are sold are blueberry, that means that \(\displaystyle 8\) of the pies sold were probably blueberry. 

Given that \(\displaystyle 10\%\) of \(\displaystyle 40\) is \(\displaystyle 4\), it follows that \(\displaystyle 20\%\) of \(\displaystyle 40\) is \(\displaystyle 8\)

Therefore, \(\displaystyle 8\) is the correct answer.

Example Question #13 : How To Find A Proportion

The ratio of rabbits to guinea pigs to hamsters in a pet store is 3:2:4. If there are 12 rabbits in the pet store, how many hamsters are there?

Possible Answers:

\(\displaystyle 12\)

\(\displaystyle 8\)

\(\displaystyle 16\)

\(\displaystyle 4\)

Correct answer:

\(\displaystyle 16\)

Explanation:

Given that the ratio of rabbits to guinea pigs to hamsters in a pet store is \(\displaystyle 3:2:4\), each number should be 4 times greater if there are 12 rabbits. This is because 3 times 4 is 12. 

Therefore, we can transform the of rabbits to guinea pigs to hamsters into \(\displaystyle 12:8:16\)

Therefore, there are 16 hamsters. 

Example Question #13 : How To Find A Proportion

Reduce the following proportion to its simplest form: 

\(\displaystyle \frac{35}{49}\)

Possible Answers:

\(\displaystyle 35\)

\(\displaystyle 5\)

\(\displaystyle 7\)

\(\displaystyle \frac{1}{5}\)

\(\displaystyle \frac{5}{7}\)

Correct answer:

\(\displaystyle \frac{5}{7}\)

Explanation:

To simplify, you find a common factor for both the numerator and denominator and divide them both by that value.  

Both \(\displaystyle 35\) and \(\displaystyle 49\) have a common factor of \(\displaystyle 7\).  

This gives us,

 \(\displaystyle \frac{35\div 7}{49\div 7}=\frac{5}{7}\).

Example Question #14 : How To Find A Proportion

Rachel and Michael earn the same amount of money per hour.  Rachel earns $64.00 for eight hours of work. How long would it take Michael to earn $96.00?

Possible Answers:

\(\displaystyle 9 hours\)

\(\displaystyle 7 hours\)

\(\displaystyle 12 hours\)

\(\displaystyle 11 hours\)

Correct answer:

\(\displaystyle 12 hours\)

Explanation:

In order to solve this problem, you would set up a direct proportion:

\(\displaystyle \frac{8}{64} =\frac{x}{96}\)

To make the process even simpler, you could first reduce the first fraction in this direct proportion. The fraction was reduced to simplest form by dividing both the numerator and the denominator by 8, the greatest common factor or the GCF.

\(\displaystyle \frac{1}{8} = \frac{x}{96}\)

\(\displaystyle 8x = 96\)

Divide both sides by the coefficient of x, which is 8:

\(\displaystyle x = 12 hours\)

Example Question #534 : Numbers And Operations

On a map, one inch is equal to \(\displaystyle 20\) miles. How many inches will be on the map for a a distance of \(\displaystyle 130\) miles?

Possible Answers:

\(\displaystyle 6\frac{1}{2}\textup{ inches}\)

\(\displaystyle 5\textup{ inches}\)

\(\displaystyle 6\textup{ inches}\)

\(\displaystyle 5\frac{1}{2}\textup{ inches}\)

\(\displaystyle 7\textup{ inches}\)

Correct answer:

\(\displaystyle 6\frac{1}{2}\textup{ inches}\)

Explanation:

The proportion for the problem will be,

\(\displaystyle \frac{1 in}{20 miles}=\frac{x}{130 miles}\).  

You then cross multiple to get 

\(\displaystyle 130=20x\).  

To find \(\displaystyle x\), you divide both sides by \(\displaystyle 20\) to get

          \(\displaystyle 6.5\)

\(\displaystyle 20)\overline{130}\)

   \(\displaystyle \underline{-120}\)

         \(\displaystyle 10\)

     \(\displaystyle \underline{-10}\)

          \(\displaystyle 0\)

Therefore

 \(\displaystyle 130 \textup{ miles}=6\frac{1}{2}\textup{ inches}\)

 

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