Word Problems: Finding Unit Rates

A wide variety of word problems are most easily solved by finding a "unit rate."  (Sometimes these types of problems can also be solved using proportions, so it can be a matter of which method you feel most comfortable with!)

Unit rate problems usually start by stating a rate for a group.  They might tell you that t-shirts cost 4 for \$15.  Or they might tell you that a carton of a dozen eggs costs \$2.40.  Or, they might tell you that a runner can run a mile (four laps) in 5 minutes. 

Then, unit rate problems will ask you to change the group.  If t-shirts are 4 for $15, how much are 3 shirts?  Or, how much is one shirt?  Or, how much would 100 shirts be?

The unit rate is the cost (or speed, or rate) for ONE item in a group.  Often finding a unit rate is the key to solving a lot of different questions that could come next. Typically, you will divide to find a unit rate, and then multiply to find the cost (or rate or speed) of a different group.

Example:

Jump rope costs $4.50 for a three yard length.  If you want 5 yards, how much will it cost?

First, find the cost per yard by dividing the total cost by the length at which the rope is sold.

$\$4.50\div3\text{ yards} =  \$1.50\text{/yard}$

(Because it is \$1.50 per ONE yard, this is the unit rate).

Now, just multiply the unit rate by the new quantity.

$1.50 \times 5 \text{yards} =  \$7.50 \text{ for } 5\text{ yards.}$

Finding unit rates can also be very helpful when you want to comparison shop.  Sometimes when items are grouped, it's hard to tell when you're getting the better price.  Find the unit rate for each group and then you can compare.

Example:

A fancy cupcake store sells 1 cupcake for \$2.00.  You can purchase half a dozen (6) cupcakes for \$10, or 20 cupcakes for $30.  How much do cupcakes cost under each price point?

First you find the unit rate of cupcakes under each price configuration:

Purchased individually: cupcakes are $2 each.

Purchased by the half dozen: cupcakes are

$\$10\div6 =  \$1.67\text{ each}$

Purchased in sets of 20 cupcakes are

$\$30\div20 =  \$1.50\text{ each}$

The unit rate of the cupcakes is clearly the lowest when you buy in packs of twenty.

Sometimes they problems will get a little more complicated and ask you something like:

If you need 19 cupcakes for a party, what is the cheapest way to get them?

You know that $30 for 20 is the cheapest way to get cupcakes, but what if you do not need all 20?  Check the other configurations to make sure you're better off buying an extra cupcake!

You know that buying in packs of 6 is cheaper than buying individually, so $$\eqalign{18 \text{ cupcakes} &=\\ 3 \text { half dozen packs} &= 3 \times \$10 = \$30\\ + 1 \text{ cupcake} &=\$30 + \$2=\$32}$$

Buying 19 cupcakes will cost \$32.  You may as well by 20 cupcakes for \$30!

Remember, finding unit rates doesn't only help with items.  It can also be used with measurements, like speed.  We usually think of speed as a unit rate (miles per hour) but sometimes you have to actually create that unit rate to solve a problem.

Example: Kareem jogs 5 miles in 75 minutes.  Julia jogs 3 miles in 30 minutes. If, after months of working out, they both improve their speed by 1 mile per minute, what is the speed of the faster runner?

First, find the unit rate (minutes per mile) for each runner (Note: we're finding minutes per mile because the division is easier, you could also choose to find miles per minute.  We prefer to keep things in whole numbers if we can!):

Kareem runs 5 miles in 75 minutes:

$75 \div 5 = 15 \text{ minutes to run one mile}$

Julia runs 3 miles in 30 minutes:

$30 \div 3 = 10 \text { minutes to run one mile}$

Next subtract a minute from each unit rate:

Kareem: $15 - 1 = 14 \text { minutes to run one mile}$

Julia: $10 - 1 = 9 \text{ minutes to run one mile}$

Compare compare how long it takes each of them to run a mile:

$14 > 9$ 

It takes Kareem longer to run a mile than Julia, so Julia has the faster speed!

Overall, when you are presented with groups and want to create new groups, or when you're supposed to comparison shop among items in groups, it's often smart to break the items down into their unit rates first.