How to find a ratio - ISEE Middle Level Quantitative Reasoning

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Question

On Katy's facebook page, she has $\small 250$ friends who are girls, $\small 150$ friends that are boys, $\small 200$ friends from her home town, and 100 friends that are from Michigan. What is the ratio of girl to friends from Michigan?

Answer

Katy has $\small 250$ friends that are girls and $\small 100$ friends from Michigan, so the ratio is $\small 250:100$ . When we divide both numbers by fifty to simplify, the ratio becomes $\small 5:2$ , since

$\small \frac{250}{50}=5$ and $\small \frac{100}{50}=2$ .

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Question

The distance between Wilsonville and Coleman is 320 miles and is represented by six inches on a map. The distance between Wilsonville and Garrett is 120 miles.

Which is the greater quantity?

(a) The distance between Coleman and Garrett on the map

(b) $5\textrm{ in}$

Answer

The closest that Coleman and Garrett can be to each other is 200 miles (if Garrett is between Wilsonville and Coleman); the farthest is 440 miles (if Wilsonville is between Coleman and Garrett).

Call the map distance between Coleman and Garrett .

The two extremes of can be calculated using proportion statements.

The minimum:

$\frac{N}{200} = \frac{6}{320}$

$\frac{N}{200} \cdot 200 = \frac{6}{320} \cdot 200$

$N = \frac{1,200}{320} = \frac{1,200\div 80}{320\div 80} = \frac{15}{4} = 3 \frac{3}{4}$

The maximum:

$\frac{N}{440} = \frac{6}{320}$

$\frac{N}{440} \cdot 440= \frac{6}{320} \cdot 440$

$N = \frac{2,640}{320} = \frac{2,640\div 80}{320\div 80} = \frac{33}{4} = 8 \frac{1}{4}$

It is therefore unclear whether the map distance is greater than or less than 5 inches.

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Question

Travis took 45 minutes to drive a total of 40 miles.

Which is the greater quantity?

(a) 55 miles per hour

(b) The average rate at which Travis drove

Answer

Travis drove 40 miles in three-fourths of an hour, so we can divide:

$40 \div \frac{3}{4} = \frac{40}{1} \times \frac{4} {3} = \frac{160} {3} = 53 \frac{1} {3}$ miles per hour,

which is less than 55.

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Question

Mike drove 120 miles to his mother's house. He finished the trip in two hours and 15 minutes. It took him one hour to drive the first 50 miles.

Which is the greater quantity?

(a) Mike's average speed over the first 50 miles

(b) Mike's average speed over the last 70 miles

Answer

We can compare miles per hour.

(a) 50 miles over a one-hour period is 50 miles per hour.

(b) 70 miles over a one-and-one-fourth-hour period is

$70 \div 1\frac{1}{4} = \frac{70}{1} \div \frac{5}{4} = \frac{70}{1} \times \frac{4}{5} = \frac{14}{1} \times \frac{4}{1} = 56$

Mike drove an average of 56 miles per hour over the last 70 miles, making (b) greater.

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Question

There are fifteen boys in a room and twelve girls. Five more girls enter. What is the ratio of girls to total students in the room after this change occurs?

Answer

After the new girls enter the room, you have boys and girls. This means that there is a total of people in the room. The ratio of girls to boys would be .

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Question

In a bowl of pieces of fruit, $\frac{2}{3}$ are apples. The rest are kiwis. If the number of apples is doubled, what is the ratio of kiwis to the total number of fruit in the newly enlarged quantity of fruit in the bowl.

Answer

We know that $\frac{2}{3}$ of the total pieces of fruit are apples. This means that there are:

$\frac{2}{3} * 45 = 30$ apples.

Thus far, we know that we must have:

apples

and

kiwis

Now, if we double the apples, we will have:

apples

and

kiwis

This means that the proportion of kiwis to total fruit will be:

or , which can be reduced to

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Question

In a given neighborhood, there are 200 vehicles. Half of these are cars, a quarter are SUVs, five percent are motorcycles, and the remaining amount are trucks. If the number of trucks are doubled, what is the ratio of motorcycles to total vehicles?

Answer

You just need to work this through step-by-step.

We know that half of the vehicles are cars; therefore, of them are cars. To find the number of SUVs, multiply by (a quarter) and get SUVs. To find the number of motorcycles, multiply by to get . Finally, there is % remaining for trucks; therefore, multiply by to get .

Now, if this is doubled, we have trucks. This means that the total number of vehicles is:

vehicles

Therefore, the ratio of motorcycles to total vehicles will be:

Reducing this, you get:

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Question

and are positive.

The ratios 20 to and to 40 are equvalent.

Which is the greater quantity?

(a)

(b)

Answer

The ratios 20 to and to 40 are equvalent, so

$\frac{20}{M} = \frac{N }{40}$

By the cross-product property,

$MN = 20 \cdot 40 = 800$

Without any futher information, however, it cannot be determined which of and is the greater. For example, and fits the condition, as does the reverse case.

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Question

and are positive.

The ratios 125 to and to 125 are equvalent.

Which is the greater quantity?

Answer

The ratios 125 to and to 125 are equvalent, so

$\frac{125}{M} = \frac{N }{125}$

By the cross-product property,

$MN = 125 \cdot 125 = 15,625$

Without any futher information, however, it cannot be determined which of and is the greater. For example, and fits the condition, as does the reverse case.

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Question

. Which of the following must be equivalent to the ratio $\frac{M}{7}$ ?

(a) $\frac{3 M}{21}$

(b) $\frac{M+ 3}{10}$

(c) $\frac{M- 3}{4}$

Answer

Two ratios are equivalent if and only if their cross products are equal. Set $\frac{M}{7}$ equal to each choice in turn and find their cross products:

(a) $\frac{M}{7} = \frac{3M}{21}$

$M \cdot 21 = 3M \cdot 7$

The cross products are equal, so regardless of the value of , the ratios are equivalent.

(b) $\frac{M}{7} = \frac{M+ 3}{10}$

$M \cdot 10 =( M+ 3 )\cdot 7$

The cross products are equal if and only if , so the ratios are not equivalent.

(c) $\frac{M}{7} = \frac{M- 3}{4}$

$M \cdot 4 =( M- 3 )\cdot 7$

The cross products are equal if and only if , so the ratios are not equivalent.

The correct response is (a) only.

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Question

$\frac{M}{6} = \frac{N}{7}$

and are positive. Which is the greater quantity?

(a)

(b)

Answer

The cross products of two equivalent fractions are themselves equivalent, so if

$\frac{M}{6} = \frac{N}{7}$

then

Multiply by 6:

$6 \cdot 6N=6 \cdot 7 M$

Since , it follows that , and by substitution,

.

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Question

In a bowl of pieces of fruit, $\frac{2}{3}$ are apples. The rest are kiwis. If the number of apples is doubled, what is the ratio of kiwis to the total number of fruit in the newly enlarged quantity of fruit in the bowl.

Answer

We know that $\frac{2}{3}$ of the total pieces of fruit are apples. This means that there are:

$\frac{2}{3} * 45 = 30$ apples.

Thus far, we know that we must have:

apples

and

kiwis

Now, if we double the apples, we will have:

apples

and

kiwis

This means that the proportion of kiwis to total fruit will be:

or , which can be reduced to

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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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Question

I have 5 white chocolates, 3 milk chocolates, and 9 dark chocolates. What is the ratio of white chocolates to the other types of chocolates?

Answer

There are 5 white chocolates. There are a total of 12 milk and dark chocolates. The ratio of white to other chocolates is 5:12.

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Question

Which ratio is equivalent to $13 \frac{1}{2} : 1 \frac{1}{2}$ ?

Answer

A ratio can be rewritten as a quotient; do this, and simplify it.

$13 \frac{1}{2} : 1 \frac{1}{2}$

Rewrite as

$13 \frac{1}{2} \div 1 \frac{1}{2} = \frac{27}{2} \div \frac{3}{2} = \frac{27}{2} \cdot \frac{2}{3} = \frac{9}{1} \cdot \frac{1}{1} = \frac{9}{1}$

or

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Question

In February, there were 12 rainy days, 6 cloudy days and 10 sunny days. What was the ratio of sunny days in February?

Answer

Identify the number of sunny days in February :

Then, creating a ratio, show the number of sunny days compared to the total number of days in the month of February (28):

Answer:

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Question

Express the following ratio in simplest form:

Answer

Rewrite this in fraction form for the sake of simplicity, and divide both numbers by :

$\frac{140}{60}=\frac{140\div 20}{60\div 20} = \frac{7}{3}$

The ratio, simplified, is .

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