How to find a proportion - ISEE Middle Level Quantitative Reasoning

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Question

The distance between Carson and Miller is 260 miles and is represented by four inches on a map. The distance between Carson and Davis is 104 miles.

Which is the greater quantity?

(a) The distance between Carson and Davis on the map

(b) $1 \frac{1}{2} \textrm{ in}$

Answer

Let be the map distance between Carson and Davis. A proportion statement can be set up relating map inches to real miles:

$\frac{N}{104} = \frac{4}{260}$

Solve for :

$\frac{N}{104} \cdot 104 = \frac{4}{260} \cdot 104$

$N = \frac{416}{260} = \frac{416\div 52 }{260\div 52} =\frac{8}{5} = 1\frac{3}{5}$

Carson and Davis are $1\frac{3}{5}$ inches apart on the map; $1\frac{3}{5} > 1 \frac{1}{2}$

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Question

The distance between Vandalia and Clark is 250 miles and is represented by six inches on a map. The distance between Vandalia and Ferrell is represented by three and three-fifths inches on a map.

Which is the greater quantity?

(a) The actual distance between Vandalia and Ferrell

(b) 150 miles

Answer

Let be the real distance between Vandalia and Ferrell. A proportion statement can be set up relating real miles to map inches:

$\frac{N}{3 \frac{3}{5}} = \frac{250}{6}$

Solve for :

$\frac{N}{3 \frac{3}{5}} \cdot 3 \frac{3}{5} = \frac{250}{6}\cdot 3 \frac{3}{5}$

$N = \frac{250}{6}\cdot \frac{18}{5} = \frac{50}{1}\cdot \frac{3}{1} = 150$

The actual distance between Vandalia and Ferrell is 150 miles.

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Question

Jay has a shelf of books, of which 60% are hardback. The rest are paperback. If 12 are hardback, how many paperbacks are there?

Answer

There are a couple different ways to solve this problem. One way is to set up an equation from the given equation. Essentially, you have to find the total number of books before you can find how many paperbacks. An equation for that could be In other works, 12 is 60% of what total amount? (Remember, in equations, we convert percentages to decimals.) Then, you would solve for x to get 20 total books. Once you know the total, you can subtract the number of hardbacks from that to get 8 paperbacks. Another way to solve this equation is to set up a proportion. That would be $\frac{60}{100}=\frac{12}{x}$ . Then, we could cross multiply to get Solving for x would again give you 20 and you would repeat the steps from above to get 8.

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Question

A given recipe calls for cups of butter for every cup of flower and cups of sugar. If you wish to triple the recipe, how many total cups of ingredients will you need?

Answer

This is an easy case of proportions. To triple the recipe, you merely need to triple each of its component parts; therefore, you will have:

cups of butter for every cup of flower and cups of sugar

Summing these up, you get:

total cups.

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Question

A witch's brew contains newt eyes for every lizard tongues. If Aurelia the witch used newt eyes in her recipe, how many lizard tongues did she need to use?

Answer

To solve this, you need to set up a proportion:

$\frac{18}{4} = \frac{x}{3}$

Multiply both sides by :

$x=\frac{54}{4}$

Simplifying, this gives you:

$\frac{27}{2}$ or lizard tongues.

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Question

Isidore could buy equally-sized blocks of cheese for . How many could he buy for ?

Answer

For this problem, set up a proportion:

$\frac{3}{48}=\frac{x}{192}$ , where represents the number of cheese blocks that Isidore can buy.

To solve this, multiply both sides by :

$\frac{192*3}{48}=x$

$\frac{576}{48}=x$

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Question

Refer to the above diagram. How many squares should be shaded in if it is desired that the fraction of the squares that are shaded in should be equivalent to the fraction of the circles that are shaded in?

Answer

There are four circles, three of which are shaded; there are eight squares. If we let be the number of squares to be shaded, then

$\frac{N}{8} = \frac{3}{4}$

$\frac{N}{8} \cdot 8 = \frac{3}{4} \cdot 8$

$N= \frac{24}{4} = 6$

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Question

$\frac{X}{Y} = \frac{14}{17}$

Which is the greater quantity?

(a) $\frac{X+Y}{Y}$

(b) $\frac{30}{17}$

Answer

From a property of proportions, if $\frac{X}{Y} = \frac{A}{B}$ , it follows that $\frac{X+Y}{Y} = \frac{A+B}{B}$ . Setting ,

$\frac{X+Y}{Y} = \frac{14+17}{17} = \frac{31}{17} > \frac{30}{17}$ .

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Question

A witch's brew contains newt eyes for every lizard tongues. If Aurelia the witch used newt eyes in her recipe, how many lizard tongues did she need to use?

Answer

To solve this, you need to set up a proportion:

$\frac{18}{4} = \frac{x}{3}$

Multiply both sides by :

$x=\frac{54}{4}$

Simplifying, this gives you:

$\frac{27}{2}$ or lizard tongues.

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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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