Ratio and 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

Cassie works a four hour shift at the coffee shop. During her shift, $\small 37$ customers order coffee, $\small 27$ order orange juice, and $\small 15$ order hot choclate. What is the ratio of customers who order orange juice to those who order coffee?

Answer

Since there are $\small 27$ customers who order orange juice, $\small 27$ is the first number of our ratio.

The second number of our ratio is $\small 37$ , because $\small 37$ people order coffee.

Therefore, our ratio is :

$\small 27:37$

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Question

Stacy has started collecting coins. She has twice as many wheat pennies as she does 1946 dimes, and three times as many 1983 quarters as she does 1946 dimes. If Stacy has $\small 4$ 1946 dimes, what is the ratio of wheat pennies to 1983 quarters?

Answer

We know that Stacy has $\small 4$ 1946 dimes.

We also know that she has twice as many wheat pennies, so we must multiply $\small 4$ by $\small 2$ to find the total number of wheat pennies.

$\small 4\cdot 2=8$

We also know that Stacy has three times as many 1986 quarters, so we multiply $\small 4$ by $\small 3$ to find the total number of 1986 quarters.

$\small 4\cdot 3=12$

Now our ratio of wheat pennies to 1986 quarters is $\small 8:12$ , but this can be simplified because both numbers are divisible by $\small 4$ .

$\small (8/4)=2$

$\small (12/4)=3$

So our ratio becomes $\small 2:3$

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Question

Steve is having a birthday party. So far, he has invited thirty-nine of his friends from school, fifteen of whom are girls. He wants to make the ratio of girls to boys at the party two to one. How many more girls does he need to invite, assuming no more boys are invited?

Answer

Counting Steve, there are forty attendees so far, fifteen of whom are girls and twenty-five of whom (including Steve) are boys. To obtain a two-to-one girl to boy ratio, he needs to invite a total of fifty girls, so he needs to invite thirty-five more.

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Question

For every 3 peanuts in a bag of trail mix, there are 1 chocolate chip and 2 raisins. What is the proportion of raisins in the mix?

Answer

Given that there are 3 peanuts for every 1 chocolate chip and 2 raisins, the proportion of raisins can be found by dividing the number of raisins by the sum of all the items in the mix. This results in:

$\frac{2}{3+1+2}=\frac{2}{6}=\frac{1}{3}$ .

Therefore, $\frac{1}{3}$ is the correct answer.

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Question

Scott's mother bakes 12 cookies and he eats half of them. Scott's mom then puts red frosting on 4 cookies and blue frosting on the rest of the cookies. What is the ratio of cookies with red frosting to cookies with blue frosting?

Answer

If Scott's mother bakes 12 cookies and he eats half of them, that means he will eat 6 cookies, leaving 6 uneaten cookies.

If Scott's mom then puts red frosting on 4 cookies and blue frosting on the rest of the cookies, 2 cookies will get blue frosting. ().

Therefore, the ratio of cookies with red frosting to cookies with blue frosting is , which is equal to .

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Question

Jose is times taller than his baby brother Michael. If Jose is inches tall, how tall is Michael?

Answer

The ratio of Jose to Michael's height is .

Thus, if Jose is inches taller than Michael must be times shorter.

We can write this as the expression, $\frac{48}{3}=16$ .

Also, an equivalent ratio for = .

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Question

Find an equivalent ratio.

Answer

An equivalent ratio of is .

This is the furthest simplification of the ratio.
The greatest common divisor is needed to simplify the expression.

Both and are divisible by , $\frac{15}{3}=5$ and $\frac{3}{3}=1$ .

Thus, the correct answer is .

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Question

Solve for .

Answer

To solve this problem the relationship between the two numbers in the first ratio needs to be established.

Since, is times larger than , an equivalent ratio must have the same relationship.

Thus, must be times larger than , and $19\cdot 3=57$

Thus,

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Question

There are students in a fifth grade class. If there are girls in the class, what is the ratio of the number of boys in the class to the total number of students in the class?

Answer

First, find the number of boys in the class by subtracting the number of girls in the class from the total number of students.

$\text{Number of boys}=36-12=24$

Now, we know that the ratio of the number of boys in the class to the total number of students in the class is $24\text{ to }36$ .

We can express that ratio as the following fraction:

$\frac{24}{36}=\frac{2}{3}$

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Question

There are students in a computer science class. If there are boys in the class, what is the ratio of the number of girls in the class to the total number of students in the class?

Answer

First, find the number of girls in the class by subtracting the number of boys in the class from the total number of students.

$\text{Number of girls}=40-15=25$

Now, we know that the ratio of the number of boys in the class to the total number of students in the class is $25\text{ to }40$ .

We can express that ratio as the following fraction:

$\frac{25}{40}=\frac{5}{8}$

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Question

In a litter of puppies, there are black puppies and white puppies. What is the ratio of the number of white puppies to black puppies?

Answer

The ratio wants us to compare the number of white puppies to the number of black puppies.

We can say the ratio of white puppies to black puppies is $8\text{ to } 4$ . When we write that as a fraction, it becomes $\frac{8}{4}=\frac{2}{1}$ , so we can also say that the ratio of white puppies to black puppies is $2\text{ to }1$ or .

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