ISEE Middle Level Quantitative : How to find a proportion

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

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Example Questions

Example Question #1 : How To Find A Proportion

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) \(\displaystyle 1 \frac{1}{2} \textrm{ in}\)

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

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

\(\displaystyle \frac{N}{104} = \frac{4}{260}\)

Solve for \(\displaystyle N\):

\(\displaystyle \frac{N}{104} \cdot 104 = \frac{4}{260} \cdot 104\)

\(\displaystyle N = \frac{416}{260} = \frac{416\div 52 }{260\div 52} =\frac{8}{5} = 1\frac{3}{5}\)

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

Example Question #2 : How To Find A Proportion

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

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

Correct answer:

(a) and (b) are equal

Explanation:

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

\(\displaystyle \frac{N}{3 \frac{3}{5}} = \frac{250}{6}\)

Solve for \(\displaystyle N\):

\(\displaystyle \frac{N}{3 \frac{3}{5}} \cdot 3 \frac{3}{5} = \frac{250}{6}\cdot 3 \frac{3}{5}\)

\(\displaystyle N = \frac{250}{6}\cdot \frac{18}{5} = \frac{50}{1}\cdot \frac{3}{1} = 150\)

The actual distance between Vandalia and Ferrell is 150 miles.

Example Question #3 : How To Find A Proportion

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

Possible Answers:

\(\displaystyle 20\)

\(\displaystyle 8\)

\(\displaystyle 12\)

\(\displaystyle 16\)

\(\displaystyle 28\)

Correct answer:

\(\displaystyle 8\)

Explanation:

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 \(\displaystyle 12=.6x.\) 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 \(\displaystyle \frac{60}{100}=\frac{12}{x}\). Then, we could cross multiply to get \(\displaystyle 60x=1200.\) Solving for x would again give you 20 and you would repeat the steps from above to get 8.

Example Question #4 : How To Find A Proportion

A given recipe calls for \(\displaystyle 2\) cups of butter for every \(\displaystyle 1\) cup of flower and \(\displaystyle 17\) cups of sugar. If you wish to triple the recipe, how many total cups of ingredients will you need?

Possible Answers:

\(\displaystyle 35\)

\(\displaystyle 24\)

\(\displaystyle 44\)

\(\displaystyle 60\)

\(\displaystyle 55\)

Correct answer:

\(\displaystyle 60\)

Explanation:

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:

\(\displaystyle 6\) cups of butter for every \(\displaystyle 3\) cup of flower and \(\displaystyle 51\) cups of sugar

Summing these up, you get:

\(\displaystyle 6+3+51 = 60\) total cups.

Example Question #871 : Isee Middle Level (Grades 7 8) Quantitative Reasoning

A witch's brew contains \(\displaystyle 4\) newt eyes for every \(\displaystyle 3\) lizard tongues. If Aurelia the witch used \(\displaystyle 18\) newt eyes in her recipe, how many lizard tongues did she need to use?

Possible Answers:

\(\displaystyle 27\)

\(\displaystyle 46\)

\(\displaystyle 13.5\)

\(\displaystyle 34\)

\(\displaystyle 54\)

Correct answer:

\(\displaystyle 13.5\)

Explanation:

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

\(\displaystyle \frac{18}{4} = \frac{x}{3}\)

Multiply both sides by \(\displaystyle 3\):

\(\displaystyle x=\frac{54}{4}\)

Simplifying, this gives you:

\(\displaystyle \frac{27}{2}\) or \(\displaystyle 13.5\) lizard tongues.

 

Example Question #211 : Numbers And Operations

Isidore could buy \(\displaystyle 3\) equally-sized blocks of cheese for \(\displaystyle \$48\). How many could he buy for \(\displaystyle \$192\)?

Possible Answers:

\(\displaystyle 10\)

\(\displaystyle 14\)

\(\displaystyle 12\)

\(\displaystyle 16\)

\(\displaystyle 9\)

Correct answer:

\(\displaystyle 12\)

Explanation:

For this problem, set up a proportion:

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

To solve this, multiply both sides by \(\displaystyle 192\):

\(\displaystyle \frac{192*3}{48}=x\)

\(\displaystyle \frac{576}{48}=x\)

\(\displaystyle 12=x\)

Example Question #7 : How To Find A Proportion

Assorted 2

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?

Possible Answers:

\(\displaystyle 4\)

\(\displaystyle 8\)

\(\displaystyle 6\)

\(\displaystyle 3\)

Correct answer:

\(\displaystyle 6\)

Explanation:

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

\(\displaystyle \frac{N}{8} = \frac{3}{4}\)

\(\displaystyle \frac{N}{8} \cdot 8 = \frac{3}{4} \cdot 8\)

\(\displaystyle N= \frac{24}{4} = 6\)

Example Question #1 : How To Find A Proportion

\(\displaystyle \frac{X}{Y} = \frac{14}{17}\)

Which is the greater quantity?

(a) \(\displaystyle \frac{X+Y}{Y}\)

(b) \(\displaystyle \frac{30}{17}\)

 

Possible Answers:

(b) is the greater quantity

It is impossible to determine which quantity is the greater from the information given

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) is the greater quantity

Explanation:

From a property of proportions, if \(\displaystyle \frac{X}{Y} = \frac{A}{B}\), it follows that \(\displaystyle \frac{X+Y}{Y} = \frac{A+B}{B}\). Setting \(\displaystyle A = 14, B = 17\),

 \(\displaystyle \frac{X+Y}{Y} = \frac{14+17}{17} = \frac{31}{17} > \frac{30}{17}\).

Example Question #9 : How To Find A Proportion

A witch's brew contains \(\displaystyle 4\) newt eyes for every \(\displaystyle 3\) lizard tongues. If Aurelia the witch used \(\displaystyle 18\) newt eyes in her recipe, how many lizard tongues did she need to use?

Possible Answers:

\(\displaystyle 27\)

\(\displaystyle 13.5\)

\(\displaystyle 46\)

\(\displaystyle 54\)

\(\displaystyle 34\)

Correct answer:

\(\displaystyle 13.5\)

Explanation:

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

\(\displaystyle \frac{18}{4} = \frac{x}{3}\)

Multiply both sides by \(\displaystyle 3\):

\(\displaystyle x=\frac{54}{4}\)

Simplifying, this gives you:

\(\displaystyle \frac{27}{2}\) or \(\displaystyle 13.5\) lizard tongues.

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