Calculating ratio and proportion - GMAT Quantitative

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Question

A furlong is a measure of length used in horse racing. Eight furlongs are equal to one mile.

Express 20 furlongs in feet.

Answer

Divide 20 furlongs by 8 furlongs per mile, then multiply by 5,280 feet per mile.

$\frac{20}{8} \times 5,280 = 2.5 \times 5,280 = 13,200$

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Question

A fathom is a unit of underwater depth equal to six feet.

Express two miles in fathoms.

Answer

Multiply 2 miles by 5,280 feet per mile, then divide by 6 feet per fathom:

$(2 \times 5,280) \div 6 = 10,560 \div 6 = 1,760 \textrm{ fathoms}$

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Question

The ratio 4 to $\frac{1}{4}$ is equal to which of the following ratios?

Answer

The ratio $\dpi{100} \small 4$ to $\frac{1}{4}$ is equal to $\frac{4}{\frac{1}{4}}$ which is $4\left ( \frac{4}{1} \right )= 16$ .

$\dpi{100} \small 16$ can be written as the ratio $\dpi{100} \small 16$ to $\dpi{100} \small 1$ .

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Question

The annual budget for a road construction project is $25,200 budgeted equally over 12 months. If by the end of the third month the actual expenses have been $7,420, how much has the construction project gone over budget?

Answer

The monthly budget is found by:

$\frac{25,200}{12}=2,100$

which for 3 months is a budget of:

$2,100\cdot 3= 6,300$

To find out how much they are over budget the budgeted amount is subtracted from the actual expenses.
$7,420 - 6,300 = 1,120$

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Question

The ratio $\dpi{100} \small 3$ to $\dpi{100} \small \frac{1}{2}$ is equal to the ratio:

Answer

The ratio $\dpi{100} \small 3$ to $\dpi{100} \small \frac{1}{2}$ is the same as $\dpi{100} \small \frac{3}{\frac{1}{2}}=3\times \frac{2}{1}=6$ ,
which equals a ratio of $\dpi{100} \small 6$ to $\dpi{100} \small 1$ .

Also, if you double both sides of the ratio, you get $\dpi{100} \small 6$ to $\dpi{100} \small 1$ .

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Question

Nishita has necklaces, bracelets, and rings in a ratio of 7:5:4. If she has 64 jewelry items total, how many bracelets does she have?

Answer

bracelets:

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Question

In a certain classroom all of the students are either sophomores or juniors. The number of boys and girls in the classroom are equal. Of the girls, $\dpi{80} \frac{3}{7}$ are sophomores, and there are 24 junior boys. If the number of junior boys in the classroom are in the same proportion to the total amount of boys as the number of sophomore girls are to the total number of girls, how many students are in the classroom?

Answer

This question seems convoluted but is actually more simple than it seems. We are told that the number of girls and boys in the classroom are equal and that $\frac{3}{7}$ of the girls are sophomores. We are then told that 24 of the boys are juniors, and that they represent a proportion of total boys equal to the proportion of sophomore girls to total girls. This means that:

$\frac{24}{x} = \frac{3}{7}$ ==> where is the total number of boys.

If we know that the number of boys and girls in the class are equal, then the total number of students in the class = 112.

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Question

A box contains red and blue marbles. The probablity of picking a red is $\frac{1}{3}$ . There are 30 blue marbles. How many total marbles are there?

Answer

If $\frac{1}{3}$ are red, then $\frac{2}{3}$ are blue, and the number of blue marbles can be written as

$blue=\frac{2}{3}total marbles$

Plug in the number of blue marbles, 30, and solve for the total marbles.

$30 = \frac{2}{3}total\rightarrow total = \frac{30}{\frac{2}{3}} =45 marbles$

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Question

On a map, one and a half inches represents sixty actual miles. In terms of , what distance in actual miles is represented by inches on the map?

Answer

Let be the number of actual miles. Then the proportion statement to be set up, with each ratio being number of actual miles to number of map inches, is:

$\frac{60}{1\tfrac{1}{2}} = \frac{x}{N}$

Simplify the left expression and solve for

$\frac{x}{N} = \frac{60}{1\tfrac{1}{2}} = 60 \div \frac{3}{2} =60 \cdot \frac{2}{3}$

$\frac{x}{N} = 40$

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Question

The Kingdom of Zenda uses an unusual currency system. It takes 16 kronkheits to make a grotnik and 12 grotniks to make a gazoo.

At current, $1 can be exchanged for 8 grotniks and 8 kronkheits. For how much American currency can a visitor from Zenda exchange a 100-gazoo bill, to the nearest cent?

Answer

$1 can be exchanged for 8 grotniks and 8 kronkheits, or, equivalently, 8.5 grotniks (8 kronkheits is one-half of a grotnik). 100 gazoos is equal to $100 \cdot 12 = 1,200$ grotniks. Therefore, if is the number of dollars that can be exchanged for the 100-gazoo bill, we can set up the proportion:

$\frac{D}{1,200} = \frac{1}{8.5}$

Solve for :

$\frac{D}{1,200} \cdot 1,200 = \frac{1}{8.5}\cdot 1,200$

$D = \frac{1,200}{8.5} \approx 141.18$

That is, the 100-gazoo bill can be exchanged for $141.18.

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Question

The Duchy of Grand Fenwick uses an unusual currency system. It takes 24 tiny fenwicks to make a big fenwick.

At current exchange rates, 1 big fenwick can be exchanged for $3.26 American currency. For how much in Grand Fenwick currency can an American tourist exchange $300 (rounded to the nearest tiny fenwick)?

Answer

$3.26 can be exchanged for 1 big fenwick, or, equivalently, 24 tiny fenwicks. We can set up a proportion statement, where is the number of tiny fenwicks for which $300 can be exchanged:

$\frac{F}{300} = \frac{24}{3.26}$

Solve for :

$\frac{F}{300} \cdot 300 = \frac{24}{3.26} \cdot 300$

$F= \frac{24\cdot 300}{3.26} \approx 2,209$

The answer is 2,209 tiny fenwicks, which can be counted up with division:

$2,209 \div 24 = 92 \textrm{ R 1}$

or 92 big fenwicks and 1 tiny fenwick.

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Question

Five different pizza places offer five different specials. Assuming that all of these pizzas are of the same thickness and that all are of the same quality, which of the following is the best buy?

Answer

Since all of the pizzas are of the same thickness and quality, to determine the best bargain, calculate the price per square inch of each. The least amount will mark the best bargain.

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A round pizza 10 inches in diameter for $5.99:

The area of the pizza in square inches is $A = \pi r^{2} = \pi \cdot5^{2} = 25\pi$

The cost per square inch: $5.99\div25\pi =0.076$ or 7.6 cents

A round pizza 12 inches in diameter for $8.99

The area of the pizza in square inches is $A = \pi r^{2} = \pi \cdot6^{2} =36\pi$

The cost per square inch: $8.99\div36\pi =0.079$ or 7.9 cents

A 9 inch by 9 inch square pizza for $6.99

The area of the pizza in square inches is $9 \cdot 9 = 81$

The cost per square inch: $6.99 \div 81 = 0.086$ or 8.6 cents

A 10 inch by 10 inch square pizza for $7.99

The area of the pizza in square inches is $10 \cdot 10 = 100$

The cost per square inch: $7.99 \div 100 = 0.080$ or 8.0 cents

A 12 inch by 8 inch rectangular pizza for $9.99:

The area of the pizza in square inches is $12 \cdot 8 = 96$

The cost per square inch: $9.99 \div 96 = 0.104$ or 10.4 cents

The round pizza 10 inches in diameter for $5.99 is the best buy.

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Question

127 people signed up for a subscription service. 45 signed up for weekly deliveries, 34 for monthly deliveries, and the rest for annual deliveries. What is the ratio of weekly subscribers to annual subscribers?

Answer

Let's first find the number of annual subscribers:

The ratio of weekly subscribers to annual subscribers is therefore .

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Question

A clothing store is having a sales for the holiday season. They mark down all items by $43\frac{4}{5}%$ . What is the sales price of a suit whose original price is ?

Answer

We start by finding the value of the mark down:

$43\frac{4}{5}=43+\frac{4}{5}=43+0.80=43.8%$

Therefore, the mark down is 43.8% or 0.438, which means that the new price of the suit is:

$340\cdot (1-0.438)=191.08$

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Question

Noah and Lilly both work for Company ABC. Lilly earns more than Noah, and each year every employee gets a salary increase. In 2012, Noah's salary was .

What was Lilly's salary in 2013?

Answer

We start by finding Lilly's salary in 2012:

$60,000 x 1.20 = $60,000 + $12,000 = $72,000

Then we account for the annual 10% increase to find Lilly's salary in 2013:

$72,000 x 1.10 = $72,000 + $7,200 = $79,200

So Lilly's salary in 2013 was $79,200.

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Question

Company sells liters of soda for , and Company sells liters of soda for .

If the amount of soda sold per dollar is the same for each company, what is ?

Answer

Since the amount of soda per dollar is the same for each company, we can equate the 2 ratios

$\frac{5liters}{2dollars} = \frac{3liters}{xdollars}$

And solve

$\frac{5}{2} = \frac{3}{x}$

(Cross multiply)

$x = \frac{6}{5} = 1.2$

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Question

Ms. Lopez earned $236.25 after working 15 hours last week. At this rate, how much will she have earned if she works 25 hours?

Answer

The first thing we must do is set up our equation. We do this by writing our earnings to hours worked ratio:

$\frac{236.25}{15\textup{ hours}}=\frac{x}{25\textup{ hours}}$

where represents her earnings after working 25 hours.

We can then solve this equation for .

$\frac{(236.25)(25)}{15}=x$

Then,

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Question

George just finished the first week of his new job. He calculated that if he works 50 hours in two weeks, his biweekly paycheck will be $812.50. If he has worked a total of 27 hours so far, how much additional money must he earn in order to reach $812.50?

Answer

He has worked 27 hours out of the 50 hours calculated. As such, we are asked how much he will earn after working the remaining 23 hours.

We can first find his hourly pay and then calculate his earnings for 23 hours.

$\frac{812.50}{50hrs}=\frac{x}{1hr}$

where represents his hourly earnings

He earns $16.25 an hour

We can now calculate how much he earns for the remaining 23 hours:

So, George needs $373.75.

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Question

A small company's workforce consists of store employees, store managers, and corporate managers in the ratio 10:3:1. How many employees are either corporate managers or store managers if the company has a total of employees?

Answer

Let be the number of store employees, the number of store managers, and the number of corporate managers.

$10x=10\times9=90$ , so the number of store employees is .

$3x=3\times9=27$ , so the number of store managers is .

, so the number of corporate managers is .

Therefore, the number of employees who are either store managers or corporate managers is .

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Question

A scale model of the newest Mars rover has a wheel radius of and its antennae are each long. If the radius of the wheel on the actual rover is , what is the length of the antennae on the actual rover?

Answer

Setting up a proportion here is the way to go. To make our proportion even easier to solve, set it up in such a way that your unknown (antennae length for the actual rover) is in the numerator. Recall that in a proportion, we want to have similar parts together. In this case, we will keep the wheel radii on the left and the antennae lengths on the right.

$\frac{45:cm}{5:cm}=\frac{x:cm}{3:cm}$

By leaving our unknown on top, we can solve this in one fell swoop. Simply multiply both sides by and simplify to get the answer.

$\frac{45:cm}{5:cm}3:cm=93:cm=27:cm$

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