GRE Quantitative Reasoning - How to find rate

If John can paint a house in hours and Jill can paint a house in hours, how long will it take for both John and Jill to paint a house together?

$\textup{1 hour and 26 minutes}$

$\textup{26 minutes}$

$\textup{24 minutes}$

$\textup{2 hours and 30 minutes}$

$\textup{6 minutes}$

Explanation

This problem states that John can paint a house in hours. That means in hour he will be able to paint $\frac{1}{2}$ of a house.

The problem also states that Jill can paint a house in hours. This means that in hour, Jill can paint $\frac{1}{5}$ of a house.

If they are painting together, you simply add the rate at which the paint separately together to find the rate at which they paint together. This means in hour, they can paint $\frac{1}{2}+\frac{1}{5}=\frac{7}{10}$ of a house. Now to find the time that paint an entire house, we simply invert that fraction, meaning that to paint an entire house together it would take them $\frac{10}{7}$ of an hour, or $\textup{1 hour and 26 minutes}$ .

The general formula for solving these work problems is $\frac{1}{\frac{1}{x}+\frac{1}{y}}$ , where is the amount of time it takes worker A to finish the job alone and is the amount of time it would take worker B to finish the job alone.

Beverly just filled up her gas tank, which has enough gas to last her, at her usual driving rate, about 45 days. However, Beverly becomes extra busy and begins driving 66.6% more than she usually does. How many days does the tank of gas last Beverly at her new rate?

10 days

15 days

30 days

33 days

27 days

Explanation

Answer: 27 days
Explanation: Recall that the rate of gas consumption is INVERSELY related to the time it takes to consume the gas. Thus, if the rate of gas consumption increases to 5/3 of it's original rate (a 66.6% increase), then the time it will take to consume all of the gas decreases to the inverse, or 3/5 of the original. The answer is thus 27 (3/5 of 45).

Two cars begin 500 miles apart and begin driving directly toward each other. One car proceeds at a rate of 50 miles per hour, while the other proceeds at a rate of 40 miles per hour. Rounding down, many minutes will it take for the two drivers to be 150 miles apart?

300

333

233

250

Explanation

You know that the distance between these cars is defined by the following equation: Answer "250" = 500 – 90t. This is because the cars get 90 miles closer every t hours. You want to solve for t when the distance is 150: 150 = 500 – 90t; –350 = –90t; t = 35/9. Recall, however, that the question asked for the rounded-down number of minutes; therefore, multiply your answer by 60: 35 * 60/9 = 2331/3

Rounding down, you get 233.

It takes Mary 45 minutes to completely frost 100 cupcakes, and it takes Benjamin 80 minutes to completely frost 110 cupcakes. How many cupcakes can they completely frost, working together, in 1 hour?

Explanation

In this rate word problem, we need to find the rates at which Mary and Bejamin frost their respective cupcakes, and then sum their respective rates per hour. In one hour Mary frosts 133 cupcakes. (Note: the question specifies COMPLETELY frosted cupcakes only, so the fractional results here will need to be rounded down to the nearest integer.) Benjamin frosts 82 cupcakes.

82 + 133=215

Ben mows the lawn in 1 hour. Kent mows the lawn in 2 hours. How long will it take them to mow the lawn working together?

1 1/2 hours

40 minutes

1 hour

45 minutes

50 minutes

Explanation

Ben mows 1 lawn in 1 hour, or 1/60 of the lawn in 1 minute. Ken mows 1 lawn in 2 hours, or 1/120 of the lawn in 1 minute. Then each minute they mow 1/60 + 1/120 = 3/120 = 1/40 of the lawn. That means the entire lawn takes 40 minutes to mow.

The cold-water faucet can fill a bucket in 30 minutes, and the hot-water faucet can fill a bucket in 60 minutes. How long will it take to fill a bucket when the two faucets are running together?

60 minutes

20 minutes

25 minutes

30 minutes

45 minutes

Explanation

The cold-water faucet fills the bucket in 30 minutes, so in 1 minute it fills 1/30 of the bucket. The hot-water faucet fills the bucket in 60 minutes, so in 1 minute it fills 1/60 of the bucket. Then, when they're both running together they fill 1/30 + 1/60 of the bucket in 1 minute.

1/30 + 1/60 = 2/60 + 1/60 = 3/60 = 1/20, so they fill the whole bucket in 20 minutes.

A Super Sweet Candy Puff Roll has 1450 calories per roll. A man eats one roll in 10 minutes. During the work day, the man eats a roll at the start of the shift and then eats another a roll every two hours after finishing the last one. Since he is watching his health, he eats only until 3 PM but will not start eating another one at any time after 3 PM. If his shift begins at 8 AM and ends at 5 PM, how many calories per minute does he consume in Super Sweet Candy Puff Rolls® during the whole work day?

9.21

14.27

5800

13.43

10.74

Explanation

It is probably easiest just to write out the eating schedule:

Roll 1: 8:00 AM - 8:10 AM

Roll 2: 10:10 AM - 10:20 AM

Roll 3: 12:20 PM - 12:30 PM

Roll 4: 2:40 PM - 2:50 PM

Therefore, he eats 4 rolls, or 1450 * 4 = 5800 calories. To get the rate, this must be divided across the whole day's minutes: 9 work hours * 60 = 540 minutes. The average calories per minute = 10.74.

Carol ate 3 pancakes in 5 minutes. If she continues to eat at the same rate, how many whole pancakes can she eat in 24 minutes?

$\dpi{100} \small 14$

$\dpi{100} \small 6$

$\dpi{100} \small 8$

$\dpi{100} \small 12$

$\dpi{100} \small 15$

Explanation

If Carol ate 3 pancakes in 5 minutes, she can eat $\dpi{100} \small \frac{3}{5}$ of a pancake every minute. $\dpi{100} \small \frac{3}{5}\ pancakes\times 24\ minutes=14.4\ pancakes$ .

That means she ate 14 whole pancakes (and an additional 2/5 of another pancake).

Quantitative Comparison

Alice has a puppy and a kitten. The puppy weighs 4 pounds and grows at a rate of 1 pound per month. The kitten weighs 2 pounds and grows at a rate of 2 pounds per month.

Quantity A: Weight of the puppy after 8 months

Quantity B: Weight of the kitten after 7 months

Quantity A is greater.

Quantity B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

Explanation

The puppy starts at 4 pounds and gains 1 pound per month for 8 months, so he weighs 4 + 8 = 12 pounds at the end of 8 months. The kitten starts at 2 pounds and gains 2 pounds per month for 7 months, so he weighs 2 + 14 = 16 pounds at the end of 7 months. Therefore Quantity B is greater.

Mario can solve $\frac{1}{y}$ problems in $n^{2}$ hours. At this rate, how many problems can he solve in $y^{2}n^{2}$ hours?