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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