Unit Prices and Speed, Distance, and Time | Grade 6 Math | LevelKite

You already know that a unit rate tells how much of one quantity goes with 1 unit of another quantity. Now you will use that idea in a store and on the road.

A unit price is a unit rate for money. It is the cost of exactly 1 unit of something, such as 1 box or 1 pound. Its label has the form money per unit, such as dollars per box.

Here is what that means. Suppose 4 notebooks cost 8 dollars, and every notebook costs the same amount. Share the 8 dollars equally among the 4 notebooks: $8 \div 4 = 2$ . Each notebook costs 2 dollars, so the unit price is 2 dollars per notebook.

In $8 \div 4 = 2$ , the 8 is the dividend (the amount you share), the 4 is the divisor (the number you divide by), and the 2 is the quotient (the result).

So when every unit costs the same amount, you find a unit price by dividing the total price by the number of units. The number of units is the divisor, and it must not be zero. You cannot share a price among zero items.

Worked example

A unit price that comes out evenly

A pack of 6 juice boxes costs 4.50 dollars. Every box costs the same amount. What is the unit price in dollars per box?

Name the two quantities. The total price is 4.50 dollars. The number of units is 6 boxes, which is not zero. You want the dollars for 1 box, so divide the dollars by the boxes.
$\text{unit price} = 4.50 \div 6$
Share the tenths first. 4.50 is the same as 45 tenths. 6 times 7 is 42, so each box gets 7 tenths. That uses 42 tenths and leaves 3 tenths still to share.
$45 \text{ tenths} - 42 \text{ tenths} = 3 \text{ tenths}$
You can split a dollar into smaller parts, so keep sharing instead of stopping with a leftover. Trade the 3 tenths for 30 hundredths. 6 times 5 is 30, so each box gets 5 hundredths and nothing is left.
$30 \text{ hundredths} \div 6 = 5 \text{ hundredths}$
Put the two parts together. Each box gets 7 tenths and 5 hundredths of a dollar, which is 0.75 of a dollar.
$4.50 \div 6 = 0.70 + 0.05 = 0.75$
Check by multiplying the quotient by the divisor. This quotient is exact (you did not round it), so the product, the result of multiplying, must equal the dividend exactly. 6 times 0.70 is 4.20, and 6 times 0.05 is 0.30.
$0.75 \times 6 = 4.20 + 0.30 = 4.50$

Answer

The unit price is $0.75$ dollars per juice box, which is 75 cents per box.

Some unit prices do not come out evenly. Suppose a 3-pound bag of carrots costs 2.00 dollars. The number of units is 3 pounds, which is not zero, so the unit price is $2.00 \div 3 = 0.666\ldots$ dollars per pound. The 6s never end.

Stores count money in cents, so round to the hundredths place. The next digit is 6, which is 5 or more, so 6 hundredths rounds up to 7 hundredths. The unit price is about $0.67$ dollars per pound.

Now multiply back: $0.67 \times 3 = 2.01$ . That is 1 cent more than $2.00$ . Nothing went wrong. You rounded the quotient up a little, so the product came out a little high.

Speed is a unit rate too. It tells how much distance something covers in 1 unit of time. A speed of 50 miles per hour means 50 miles of distance for each 1 hour of time.

In this lesson, every speed is a constant speed. That means the speed stays the same for the whole trip, so equal amounts of time always cover equal distances. The table shows a car that keeps a constant speed of 50 miles per hour.

A car at a constant speed of 50 miles per hour

Time (hours)	Speed times time	Distance (miles)
$1$	$50 \times 1$	$50$
$2$	$50 \times 2$	$100$
$3$	$50 \times 3$	$150$
$3.5$	$50 \times 3.5$	$175$

The table follows one pattern. Before you write it with symbols, name each quantity. Let $d$ be the distance, let $t$ be the time, and let $r$ be the constant speed ( $r$ for rate). The units must fit together: if $r$ is in miles per hour, then $t$ is in hours and $d$ is in miles.

Each hour adds $r$ miles, so after $t$ hours the distance is $d = r \times t$ .

Multiplication and division are inverse operations: each one undoes the other. In the table, $150 = 50 \times 3$ . Dividing 150 by 3 undoes the times 3 and brings back the speed: $150 \div 3 = 50$ . Dividing 150 by 50 undoes the times 50 and brings back the time: $150 \div 50 = 3$ .

The same works with the letters. Dividing $d$ by $t$ leaves $r$ , as long as $t$ is not zero. Dividing $d$ by $r$ leaves $t$ , as long as $r$ is not zero. So the three forms in the next table describe one relationship. Use the one that has the quantity you want alone on one side.

Three forms of one relationship, at a constant speed

You want	You know	Calculate	Units, for miles and hours
Distance $d$	Speed $r$ and time $t$	$d = r \times t$	miles per hour times hours gives miles
Speed $r$	Distance $d$ and time $t$ , where $t$ is not zero	$r = d \div t$	miles divided by hours gives miles per hour
Time $t$	Distance $d$ and speed $r$ , where $r$ is not zero	$t = d \div r$	miles divided by miles per hour gives hours

Key concept

A unit rate links a total to a number of units

A unit price and a constant speed are both unit rates: the amount that goes with 1 unit. When every unit gets the same amount, the total equals the unit rate times the number of units. For a speed, the total is the distance and the units are units of time. Division undoes multiplication, so this one relationship answers three questions:

How much for 1 unit? Divide the total by the number of units. Unit price is total price divided by number of units. Speed is distance divided by time.
How much in all? Multiply the unit rate by the number of units. Distance is speed times time.
How many units? Divide the total by the unit rate. Time is distance divided by speed.

Every number you divide by must be nonzero.

\text{total} = \text{unit rate} \times \text{number of units}

Worked example

Finding a speed

A cyclist rides 45 miles in 2.5 hours at a constant speed. What is her speed in miles per hour?

Name the quantities. The distance is d = 45 miles. The time is t = 2.5 hours, which is not zero. Speed is the distance for 1 hour, so divide the miles by the hours.
$r = d \div t = 45 \div 2.5$
Make the divisor a whole number. Multiply both numbers by the same nonzero number, 10. Just as many 25s fit into 450 as 2.5s fit into 45, so the quotient stays the same.
$45 \div 2.5 = 450 \div 25$
Divide. 25 times 10 is 250, which leaves 200. 25 times 8 is 200, which leaves nothing. So 25 fits 10 + 8 = 18 times.
$450 \div 25 = 18$
Check. The quotient is exact, so the speed times the time must equal the distance exactly. 18 times 2 is 36, and 18 times 0.5 is 9.
$18 \times 2.5 = 36 + 9 = 45$

Answer

The cyclist's speed is $18$ miles per hour.

Worked example

Finding a distance and a time

A train travels at a constant speed of 80 kilometers per hour. (a) How far does it travel in 3.5 hours? (b) How long does it take to travel 200 kilometers?

Part (a). The speed is r = 80 kilometers per hour and the time is t = 3.5 hours. Both use hours, so the units fit and the distance comes out in kilometers. Each hour adds 80 kilometers, so multiply the speed by the time. 80 times 3 is 240, and 80 times 0.5 is 40.
$d = r \times t = 80 \times 3.5 = 240 + 40 = 280$
Part (b). Now the distance is d = 200 kilometers and the time t is what you want. You know that 80 times t equals 200. Division undoes multiplication, so divide the distance by the speed. The speed, 80, is not zero.
$t = d \div r = 200 \div 80$
Divide. 80 times 2 is 160, which leaves 40 kilometers. You can split time into parts, so keep going instead of writing a remainder. 40 is half of 80, so the last 40 kilometers take 0.5 hour.
$200 \div 80 = 2 + 0.5 = 2.5$
Check part (b). The quotient is exact, so the speed times the time must equal the distance exactly.
$80 \times 2.5 = 160 + 40 = 200$

Answer

The train travels $280$ kilometers in $3.5$ hours. It takes $2.5$ hours, or two and a half hours, to travel $200$ kilometers.

Here is the idea to keep. A unit price and a constant speed are both unit rates: the amount for 1 unit. Divide a total by a nonzero number of units to find the unit rate. Multiply the unit rate by the number of units to find a total. Divide a total by a nonzero unit rate to find the number of units. Always label the answer with its unit, such as dollars per box or miles per hour.

You can split prices, distances and times into parts, so write a decimal quotient, not a remainder. Then multiply back to check. An exact quotient returns the total exactly. A rounded quotient returns a number close to it.

Next, you will use unit rates to solve word problems, compare unit rates, and find mistakes in unit rate work.