Ratio and Proportion - ISEE Middle Level Math

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Question

Phil earns for each hour he works. For every hour he works, he then gives to his sister Lola. How much money will Lola have if Phil works hours?

Answer

To Solve:

Multiply the Lola receives by the hours Phil worked:

$\small $2.00 \cdot 13.5=27$

Phil will give Lola if he works hours.

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Question

For every $3 I earn at work, I donate $1 to charity. How much money will I donate if I make $27.00/week.

Answer

To find the amount of the donation, divide 27 by 3.

$27\div 3=9$

The answer is 9.

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Question

Read this problem, but do not solve it: Two inches on a map represent twenty-five miles of actual distance. If Pierce Springs and Buchanan Falls are eight inches apart on the map, how far apart are they in actuality?

If we let be the actual distance between Pierce Springs and Buchanan Falls, which proportion could be used to solve this problem?

Answer

The ratios that are set equal to each other in a proportion must compare the same quantities in the same order.

In each ratio, we can put number of map inches in the numerator and the number of actual miles in the denominator.

One ratio is two map inches to twenty-five actual miles (the map scale); this ratio is $\frac{2}{25}$ .

The other ratio is eight map inches to actual miles (the distance between Pierce Springs and Buchanan Falls); this ratio is $\frac{8}{N}$ .

The proportion statement that sets these equal to each other is $\frac{8}{N} = \frac{2}{25}$ , and is therefore the correct choice.

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Question

A Spanish class has $\small 14$ seniors and $\small 13$ juniors. What proportion of the class is juniors?

Answer

A proportion is an amount that is part of a whole. There are $\small 27$ students in the class in total. This question asks for the proportion that are juniors. There are $\small 13$ juniors out of $\small 27$ students, therefore the proportion is:

$\frac{13}{27}$

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Question

The distance between Youngston and Wynne is 240 miles in reality and three inches on a map. On the same map, Charlesville and Petersburg are one and three-fourths inches apart. How far apart are they in reality?

Answer

240 real miles is represented by 3 map inches, making this a ratio of $240 \div 3 = 80$ real miles per map inch.

Therefore, one and three-fourths inches represents

$80 \times 1 \frac{3}{4} = \frac{80}{1} \times \frac{7}{4} = \frac{20}{1} \times \frac{7}{1} = 140$ miles, the distance between Charlesville and Petersburg.

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Question

What is the value of x in $\frac{x+4}{x}=\frac{4}{12}?$

Answer

Since this is a proportion, you can cross-multiply. Once you do that, the left side is Your right side is . Set those equal to each other. Then, combine like terms. Subtract from both sides so that the equation is now . Divide both sides by Your answer is

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Question

If the ratio of boys to girls in a classroom is , and there are a total of of students in the classroom, how many boys are in the classroom?

Answer

If the ratio of boys to girls in a classroom is , that means that there are boys for every girls. Thus, when there are students in a classroom, the breakdown will be boys and girls. If there are students in a classroom, the breakdown will be boys and girls, which translates to a ratio of , or .

Thus, if there are students, will be boys.

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Question

Robert, Jeff, and Paul are sharing a bag of chips that contains 20 chips. The three of them eat all of the chips. If Robert has eaten 8 chips, and Jeff eats twice as many chips as Paul, how many chips has Jeff eaten?

Answer

What do we know? We know that there are 20 chips in the bag, and we know that Robert has eaten 8 of them. Thus, we can calculate that there are chips remaining. Of this remaining, Jeff has eaten 2 parts and Paul has eaten 1 part: that's 3 parts, so let's calculate how many chips constitute each part:

$\frac{12}{3}=4$

So, each part is equal to 4 chips.

Jeff has eaten 2 parts, so gives us our answer.

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Question

Michelle is having a party, and she is experimenting with different mixtures of soda, trying to come up with something original. In particular, she likes a drink she made when she mixed together three ounces of cola and five ounces of grape soda. She has two and a half liters of cola and wants to use it all to make some of this drink; how much grape soda does she need to mix it with?

Answer

The ratio of ounces of cola to ounces of grape soda in the initial mixture can be expressed as $\frac{3}{5}$ . It must be equal to that of liters of cola to liters of grape soda in the mixture Michelle will make for the party, which, since the number of liters of grape soda is unknown, is $\frac{2\frac{1}{2}}{N}$ . Set these equal and solve for :

$\frac{2\frac{1}{2}}{N} = \frac{3}{5}$

Set the cross-products equal to each other:

$3 N = 2\frac{1}{2} \cdot 5$

$3 N = \frac{5}{2} \cdot 5$

$3 N = \frac{25}{2}$

$\frac{1}{3} \cdot 3 N =\frac{1}{3} \cdot \frac{25}{2}$

$N = \frac{25}{6} = 4 \frac{1}{6}$

Michelle will use $4 \frac{1}{6}$ liters of grape soda in the final mixture.

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Question

If one cupcake costs 75 cents, how much does a dozen cupcakes cost?

Answer

You can solve this problem using a proportion.

$\frac{1}{0.75}=\frac{12}{x}$

To solve for x, cross multiply.

$1\times x=12\times.75$

9 dollars

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Question

If you can purchase two pairs of jeans for 50 dollars, how many pairs of jeans can you purchase with 200 dollars?

Answer

You can use a proportion to solve this problem.

$\frac{2}{50}=\frac{x}{200}$

Cross multiply to solve for x.

$50\times x=200\times2$

$400\div50 = 8$

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Question

You survey students about the plans for a new practice court for the basketball team and they must give a yes or no answer. students are against the use of funds on this project. What proportion of students said that they agreed with the plan?

Answer

If disagreed with the plan, that means the rest agreed.

So to find that value, you must do the total minus the no's

.

To find the proportion you must divide the part by the whole giving us an answer of

.

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Question

If Mike gets two times the amount of work done than Joe, what is a ratio representing the work that each get done?

Answer

Since Mike does twice the work that Joe does, his part of the ratio must be double the value of Joe's. The only ratio given that has that is .

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Question

$\frac{A}{B} = \frac{3}{4}$

$\frac{A +B}{B} = ?$

Answer

By a proportion property, if $\frac{A}{B} = \frac{C}{D}$ , it follows that $\frac{A+B}{B} = \frac{C+ D}{D}$ .

Setting ,

if $\frac{A}{B} = \frac{3}{4}$ , then $\frac{A +B}{B} = \frac{3+4}{4} = \frac{7}{4}$ .

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Question

$\frac{A}{B} = \frac{5}{7}$

Answer

The cross-products (the product of a numerator and the other denominator) of a proportion are equal, so, if

$\frac{A}{B} = \frac{5}{7}$ , then

.

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Question

Which statement does not follow from the statement $\frac{A}{9} = \frac{B }{11}$ ?

Answer

If $\frac{A}{9} = \frac{B }{11}$ , then:

The ratio of the numerators is equivalent to that of the denominators, in the same order. Hence,

$\frac{A}{B} = \frac{9} {11}$ and $\frac{9} {A}= \frac{11}{B }$ are true.

The reciprocals of the ratios are equivalent, so

$\frac {9}{A}= \frac{11}{B }$

However, it does not hold that

$\frac{A}{11} = \frac{B }{9}$ .

This can be seen as follows:

$\frac{A}{9} = \frac{B }{11}$

$\frac{A}{9} \cdot \frac{9}{11}= \frac{B }{11}\cdot \frac{9}{11}$

$\frac{A}{11}= \frac{9B }{121} e \frac{B}{9}$ .

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Question

What is the least common multiple of 5, 12, and 15?

Answer

You can write out the multiples of each number, and find the first number that is common to all of them.

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60

12: 12, 24, 36, 48, 60

15: 15, 30, 45, 60

The correct answer is 60.

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Question

What is the greatest common factor of 12, 24 and 96?

Answer

To find the greatest common factor of three numbers, first factor each number as a product of prime numbers:

$\dpi{100} 96: 2\times 2\times 2\times 2\times 2\times 3$

$\dpi{100} 24: 2\times 2\times 2\times 3$

$\dpi{100} 12: 2\times 2\times 3$

Now take all of the factors that are in common between 12, 24 and 96. In this case, that is 2, 2 and 3.

$\dpi{100} 2\times 2\times 3=12$

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Question

Logan needs to answer 60 questions to finish his test. If he has answered 25% of the questions in 20 minutes, how long will it take him to complete the entire test?

Answer

To solve:

First, find the number of answers completed by calculating:

$\small 60\cdot 0.25=15$

Then, find how long it would take him to complete the 60 question test if for every 20 minutes he answer 15 questions:

$\small 60 \div 15=4$ (sets of 15 questions)

$\small \small \small 4\cdot 20 =80$ minutes altogether to finish the test.

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Question

A car travels miles in hours. At what rate does the car travel?

Answer

Divide the number of miles by the number of hours to get the number of miles per hour.

$\textup {rate}=\frac{\textup {distance}}{\textup {time}}=\frac{\textup {miles}}{\textup {hours}}$

$\frac{684\textup { miles}}{12\textup { hours}} = 57\textup { miles per hour}$

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