# Distance and Work: Rate Formula

## Formulas

A rate is how fast something is growing, changing, or being performed. The overarching rate formula is:

Amount = Rate \times Time

When the rate is a speed, this simplifies to the familiar formula:

Distance = Speed \times Time

In questions about speed, especially where an object travels at one speed for a while, then at another speed, keep in mind that you never find the numerical average of two different speeds. If the question ask for average velocity for the whole trip, then you add the distances from both parts of the trip to find the total distance, and add the times of both parts of the trip to find the total time, and use those and the formula above to calculate the speed.

When the rate is a rate of work being done, then when two people work together, their combined rate is the sum of their respective individual rates. Make sure what are you adding are the rates, not anything else.

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## Practice Questions

**Q1.** A car drives 40 miles on local roads at 20 mph, and 180 miles on the highway at 60 mph, what is the average speed of the entire trip?

(A) 36 mph

(B) 40 mph

(C) 44 mph

(D) 52 mph

(E) 58 mph

**Q2**. When Mary paints a house, it takes her 4 hours. When Lisa joins Mary, and they work together, it takes them only 3 hours to paint a house of the same size. How long would it take for Lisa to paint a house of the same size by herself?

(A) 5 hr

(B) 6 hr

(C) 7 hr

(D) 12 hr

(E) 20 hr

## Answers and Explanations

**Q1**. In phase #1 of the trip, the car traveled 40 mi at 20 mph. That time of this phase was:

time = \dfrac{distance}{rate} = \dfrac{(40 mi)}{(20 mph)} = 2 hr

In phase #2 of the trip, the car traveled 180 mi at 60 mph. That time of this phase was:

time = \dfrac{distance}{rate} = \dfrac{(180 mi)}{(60 mph)} = 3 hr

The total distance of the trip = 40 mi + 180 mi = 220 mi

The total time of the trip = 2 hr + 3 hr = 5 hr

The average speed of trip is given by

speed = \dfrac{distance}{time} = \dfrac{(220 mi)}{(5 hr)} = 44 mph

**Answer = C**

**Q2**. Here, the rate equation becomes:
(\# \space of \space houses) = (painting \space rate) \times (time)

When Mary paints a house, it takes her 4 hours. Thus

(1 \space house) = (Mary's \space rate) \times (4 \space hr)

so her rate is \dfrac{1}{4}.

When Mary & Lisa paint together, it takes 3 hrs. Thus

(1 \space house) = (combined \space rate) \times (3 \space hr)

and the combined rate = \dfrac{1}{3}.
To find a combined rate, we add individual rates.

(combined \space rate) = (Mary's \space rate) + (Lisa's \space rate)

\dfrac{1}{3} = \dfrac{1}{4} + (Lisa's \space rate)

(Lisa's rate) = \dfrac{1}{3}-\dfrac{1}{4} =\dfrac{1}{3} \times \dfrac{4}{4}-\dfrac{1}{4} \times \dfrac{3}{3} =\dfrac{4}{12}-\dfrac{3}{12} =\dfrac{1}{12}

Lisa’s rate is \dfrac{1}{12} of a house every hour, or in other words, 1 house in 12 hrs. Thus, it would take her 12 hours to paint a house of the same size.

**Answer = D**