How to find a proportion - ISEE Lower Level Quantitative Reasoning

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Question

A recipe calls for $\dpi{100} \frac{2}{3}$ of a cup of flour and $\dpi{100} \frac{2}{3}$ of a cup of sugar. How many cups of sugar and flour combined does the recipe call for?

Answer

Add $\dpi{100} \frac{2}{3}+\frac{2}{3}$ .

Since both of these fractions have the same denominator, we just rewrite 3 on the bottom. When you add fractions, then add straight across the top, $\dpi{100} 2+2=4$ .

This gives us $\dpi{100} \frac{4}{3}$ .

We can turn this into a mixed number by dividing $\dpi{100} 4\div 3=1$ with a remainder of 1. We place the orginal one as our whole number, and the remainder over the 3 for the fraction, $\dpi{100} 1\frac{1}{3}$ .

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Question

Which fraction is the greatest?

Answer

As the denominator gets bigger, the fractions gets smaller, so when looking for the largest fraction, we are looking for one with the highest numerator and the lowest denominator. In this case, $\small \frac{2}{3}$ has the largest numerator and the smallest denominator.

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Question

A foot tree casts a shadow of feet. How long is the shadow of a nearby foot tall building if the tree and its shadow are in the same ratio as the building and its shadow?

Answer

To solve this, I need to set up a proportion, and then solve algebraically.

feet, the height of the tree, is equivalent in our proportion to feet, the building's height.

feet, the tree's shadow, is equivalent in our proportion to , the building's shadow.

$\frac{30}{20}=\frac{300}{x}$

Cross-multiply:

Divide each side by :

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Question

Ralph watched $\frac{1}{16}$ of a movie that is 2 hours and 8 minutes long. How much of the movie did he watch (in minutes)?

Answer

The first step is to convert the length of the movie from hours to minutes. A movie that is 2 hours and 8 minutes long is 128 minutes long.

128 divided by 16 is 8. Therefore, 8 minutes is $\frac{1}{16}$ of the movie.

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Question

There are 18 tea bags in a box. Beth uses 2 tea bags, Brad uses 1 tea bag, and Joe uses 1 tea bag. What fraction of the tea bags now remain?

Answer

If there are 18 tea bags in a box and Beth uses 2 tea bags, Brad uses 1 tea bag, and Joe uses 1 tea bag, then 14 tea bags will remain.

Given that $\frac{14}{18}=\frac{7}{9}$ , the correct answer is $\frac{7}{9}$ .

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Question

If Katie's computer has 60% of its power left after being on for 2 hours, how many hours of battery power remain?

Answer

If Katie's computer has 60% of its power left after being on for 2 hours, then that means 40% of the battery was used to power the 2 hours. X represents the total number of hours that the battery can last.

$\frac{40}{100}=\frac{2}{x}$

Next, we divide each side of the equation by 40. This results in:

.

Since 60% of 5 is equal to 3, the correct answer is 3.

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Question

Marcus is 9 years old. He has been playing soccer for 3 years. If he plays soccer for a 4th year, for what proportion of his life will he have played soccer?

Answer

If Marcus is 9 years old and has been playing soccer for 3 years, he will be 10 years old during his 4th year. He has therefore played soccer for 4 out of 10 years, or

$\frac{4}{10}=\frac{2}{5}$ of his life.

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Question

If 5 out of 20 students in a class have green eyes, what percentage of students have green eyes?

Answer

Percentages are fractions out of 100, so the way to solve the problem is to have the denominator be 100. In order to do this we must multiply 20 by 5 in order to get 100.

If we multiply the denominator by 5, we must do the same to the numerator.

When we do this, we are left with $\small \frac{25}{100}$ which is 25%

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Question

If Katie's computer has of its power left after being on for hours, how many hours of battery power remain?

Answer

If Katie's computer has 60% of its power left after being on for 2 hours, then that means 40% of the battery was used to power the 2 hours. X represents the total number of hours that the battery can last.

$\frac{40}{100}=\frac{2}{x}$

Next, we divide each side of the equation by 40. This results in:

.

Since 60% of 5 is equal to 3, the correct answer is 3.

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Question

A cookie recipe that bakes 12 cookies calls for 3 eggs. How many eggs would be needed to make 36 cookies?

Answer

36 cookies is 3 times more than 12. Multiply the number of eggs by 3 to get 9.

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Question

A group of 6th grade students at Manor Intermediate are going on a field trip to the local theme park.

There are 2 chaperones required for every 9 students on the field trip. How many chaperones will need to attend the field trip if there are 27 students going?

Answer

The best way to solve this problem is to set it up as two ratios that are equivalent to each other. This forms a proportion. Make sure to set up the ratios so that the relationships go together across the equal sign.

$\frac{2}{9}= \frac{\square }{27}$

Use cross products to then solve for the missing value.

$2\times 27= 54$

Therefore, mental math or division can be used to find

$9\times \square = 54$ .

Since $9\times 6= 54$ ,

there would need to be 6 chaperones attending the field trip.

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Question

Give the value of that makes this proportion statement correct:

$\frac{49}{N} = \frac{14}{24}$

Answer

Cross-multiply, then solve for :

$\frac{49}{N} = \frac{14}{24}$

$14 N = 49 \cdot 24$

$14 N \div 14= 49 \cdot 24\div 14$

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Question

Give the value of that makes this proportion statement correct:

$\frac{N}{60} = \frac{14}{24}$

Answer

Solve for by multiplying both sides by 60:

$\frac{N}{60} = \frac{14}{24}$

$\frac{N}{60} \cdot 60= \frac{14}{24}\cdot 60$

$N = \frac{14}{24}\cdot \frac{60}{1} = \frac{840}{24} = 35$

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Question

If Eric can read a page from his text book in minutes, how many pages can he read in 15 minutes?

Answer

The problem can be represented by the proportion $\frac{1}{p}=\frac{x}{15}$ .

Solve: $x=\frac{15}{p}$

Eric can read $\frac{15}{p}$ pages in 15 minutes.

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Question

If Kristin makes $300 in a week when she works 15 hours, how much would she make if she worked 24 hours next week at the same rate?

Answer

To find the rate at which she is currently paid, divide $300 by the hours she works, 15, to see that she earns $20 per hour. To find her earnings for next week, multiply 24 hours by $20 per hour to see that she will make $480 next week.

Or set up the proportion $\frac{300}{15}=\frac{x}{24}$ and solve for .

$\frac{24*300}{15}=x$

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Question

If 1 inch on a map represents 5 miles, how far apart, in miles, are two cities that are 6.6 inches apart on the map?

Answer

Set up a proportion with the inches on the map and physical miles:

$\frac{1}{5}=\frac{6.6}{x}$

Solve for x:

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Question

A bag of marbles contains 4 green marbles, 7 black marbles, and 5 red marbles. What is the proportion of green marbles?

Answer

First, sum the total number of marbles:

Next, divide the total number of green marbles by the total number of marbles and simplify:

$\frac{4}{16}=\frac{1}{4}$

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Question

A bag has 11 green chips, 7 blue chips, and 6 red chips. What is the proportion of green chips?

Answer

The proportion is determined by dividing the number of selected items by the total number of items.

$P=\frac{green}{total}=\frac{11}{11+7+6}=\frac{11}{24}$

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Question

A drawer contains 5 pairs of blue socks, 2 pairs of white socks, 4 pairs of brown socks, and 7 pairs of black socks. What is the proportion of brown socks?

Answer

A proportion is determined by dividing the number of a specific event by the total number of events:

$P=\frac{4}{5+2+4+7}=\frac{4}{18}=\frac{2}{9}$

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Question

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

Answer

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:

$\frac{6}{24}$

We can reduce the fraction by removing a common factor.

$\frac{6}{24}=\frac{6\div6}{24\div6}=\frac{1}{4}$

The percentage that is equivalent to $\frac{1}{4}$ is 25 percent, or .

Therefore, is the correct answer.

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