Unit Prices and Multi-Step Decimal Problems | Grade 5 Math | LevelKite

Stores often sell things in groups: a pack of juice boxes, a bag of apples, a box of pencils. The price tag gives the cost of the whole group. But what does just one cost?

A unit is one of the things you are counting or measuring, such as one juice box or one pound of apples. The unit price is the cost of one unit.

In this lesson, every price is written as a decimal number of dollars. Think about a pack of $4$ erasers that costs $2.00$ dollars. Each eraser costs the same amount, so the $2.00$ dollars is shared equally among the $4$ erasers. Sharing equally means dividing: $2.00 \div 4 = 0.50$ . One eraser costs $0.50$ dollars, which is $50$ cents. That is the unit price of an eraser.

Key concept

The unit price is the cost of one unit

When every unit in a group costs the same amount, divide the total cost by the number of units. The result is the unit price: the cost of one unit.

You can multiply the unit price by a number of units to find what that many cost. This works as long as each of those units is sold at that same unit price.

Unit prices also let you compare. When two packs sell the same kind of unit, the pack with the lower unit price charges less for each unit.

\text{unit price} = \text{total cost} \div \text{number of units}

Worked example

Find a unit price and use it

A pack of $6$ juice boxes costs $4.50$ dollars. Each box costs the same amount. At that same price for each box, how much would $10$ boxes cost?

First find the unit price. The total cost is shared equally among the 6 boxes, so divide the total cost by the number of boxes. It can help to think of 4.50 dollars as 450 cents: 450 cents shared among 6 boxes is 75 cents for each box.
$4.50 \div 6 = 0.75$
Say what the result stands for. The 0.75 is the unit price: 1 juice box costs 0.75 dollars. Check it by multiplying back: 6 boxes at 0.75 dollars each must cost the same as the whole pack, and they do.
$6 \times 0.75 = 4.50$
Now find the cost of 10 boxes. Each box costs 0.75 dollars, so 10 boxes cost 10 groups of 0.75 dollars. Multiply the number of boxes by the unit price. The result, 7.50, stands for the dollars that 10 boxes cost.
$10 \times 0.75 = 7.50$

Answer

At the same unit price, $10$ juice boxes cost $7.50$ dollars.

With the cost of 1 juice box, you can find the cost of other numbers of boxes at that same price

Number of boxes	In words	Number sentence	Cost in dollars
$1$	The cost of the pack shared equally among $6$ boxes	$4.50 \div 6 = 0.75$	$0.75$
$3$	$3$ boxes at $0.75$ dollars each	$3 \times 0.75 = 2.25$	$2.25$
$6$	$6$ boxes at $0.75$ dollars each	$6 \times 0.75 = 4.50$	$4.50$
$10$	$10$ boxes at $0.75$ dollars each	$10 \times 0.75 = 7.50$	$7.50$

An offer is a price for a certain amount, such as $8$ bars for $5.20$ dollars. Suppose two offers sell the same kind of thing in different amounts. Looking only at the two price tags can fool you, because the tags are for different amounts.

To compare fairly, find what each offer charges for the same amount: one unit. That is its unit price. The offer with the lower unit price charges less for each unit.

This comparison makes sense only when both unit prices are for the same kind of unit, such as one bar and one bar, or one pound and one pound. Pack size alone does not tell you which offer has the lower unit price. Sometimes it is the bigger pack, and sometimes it is the smaller one. Find each unit price, then compare.

Worked example

Compare two offers

A store sells the same granola bars in two packs. Pack A has $8$ bars and costs $5.20$ dollars. Pack B has $12$ bars and costs $7.20$ dollars. Which pack has the lower unit price? How much less does one bar cost in that pack?

Find the unit price of Pack A. Its cost is shared equally among its 8 bars, so divide the total cost by the number of bars. The result stands for the cost of 1 bar in Pack A: 0.65 dollars.
$5.20 \div 8 = 0.65$
Find the unit price of Pack B the same way. The result stands for the cost of 1 bar in Pack B: 0.60 dollars.
$7.20 \div 12 = 0.60$
Compare the two unit prices. Both are the cost of 1 bar of the same kind, so the comparison is fair. Both have 6 tenths, and 0 hundredths is less than 5 hundredths, so 0.60 is less than 0.65. Pack B has the lower unit price.
$0.60 < 0.65$
Subtract the lower unit price from the higher one. The result, 0.05, stands for how many dollars less 1 bar costs in Pack B. That is 5 cents.
$0.65 - 0.60 = 0.05$

Answer

Pack B has the lower unit price. One bar costs $0.05$ dollars less in Pack B, which is $5$ cents less, even though Pack B has the higher total cost.

Some problems cannot be answered with one calculation. A multi-step problem needs two or more calculations to reach the answer. Often the result of one step is a number you use in a later step.

The tricky part is keeping track of what each number means. So give every step a name. Before you calculate, say what the step will find. After you calculate, say what the result stands for.

Try it on a small problem. Mia buys $3$ notebooks that cost $2.75$ dollars each. She pays with $10.00$ dollars. How much change does she get? The change is the money Mia pays minus the money she spends. You do not know yet what she spends, so that is the first thing to find. The table names both steps.

Mia's two steps. The last result answers the question: Mia gets 1.75 dollars in change.

Step	What the step finds	Number sentence	What the result stands for
$1$	The cost of $3$ notebooks at $2.75$ dollars each	$3 \times 2.75 = 8.25$	$8.25$ dollars: what Mia spends
$2$	The change: money paid minus money spent	$10.00 - 8.25 = 1.75$	$1.75$ dollars: what Mia gets back

Worked example

A problem with four calculations

At a fruit stand, $4$ pounds of apples cost $5.00$ dollars. Leo buys $7$ pounds of apples at that same price for each pound. He also buys a jar of honey for $6.45$ dollars. He pays with $20.00$ dollars. How much change does Leo get?

Plan first. The question asks for Leo's change. You can reach it with four calculations: the cost of 1 pound of apples, then the cost of 7 pounds, then the total Leo spends, and last his change. In this plan, each result is used in the next calculation.
Find the cost of 1 pound of apples. You know that 4 pounds cost 5.00 dollars, and each pound costs the same, so divide the total cost by the number of pounds. The result, 1.25, stands for the unit price: 1.25 dollars for 1 pound.
$5.00 \div 4 = 1.25$
Find the cost of Leo's apples. He buys 7 pounds at 1.25 dollars for each pound, so multiply. The result, 8.75, stands for the dollars Leo pays for 7 pounds of apples.
$7 \times 1.25 = 8.75$
Find the total Leo spends. He buys the apples and the honey, so add the two costs. The result, 15.20, stands for the dollars Leo spends in all.
$8.75 + 6.45 = 15.20$
Find the change. Subtract the total Leo spends from the money he pays. The result, 4.80, stands for the dollars Leo gets back.
$20.00 - 15.20 = 4.80$

Answer

Leo gets $4.80$ dollars in change.

Here is the big idea. A unit price is the cost of one unit. When every unit costs the same amount, you find it by dividing the total cost by the number of units. With it, you can find the cost of other amounts at that same price, and you can compare two offers for the same kind of unit.

In a multi-step problem, name each step and say what its result stands for. Then you know where each number belongs and when you have answered the question.

Next, you will learn to check whether a decimal answer is reasonable and to find and fix mistakes in a multi-step solution.