27.2: Adding and Subtracting Decimals with Few Non-Zero Digits
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Lesson
Let's add and subtract decimals.
Exercise 27.2.1: Do the Zeros Matter?
- Evaluate mentally: 1.009+0.391
- Decide if each equation is true or false. Be prepared to explain your reasoning.
- 34.56000=34.56
- 25=25.0
- 2.405=2.45
Exercise 27.2.2: Calculating Sums
- Andre and Jada drew base-ten diagrams to represent 0.007+0.004. Andre drew 11 small rectangles. Jada drew only two figures: a square and a small rectangle.

- If both students represented the sum correctly, what value does each small rectangle represent? What value does each square represent?
- Draw or describe a diagram that could represent the sum 0.008+0.07.
- Here are two calculations of 0.2+0.05. Which is correct? Explain why one is correct and the other is incorrect.

- Compute each sum. If you get stuck, consider drawing base-ten diagrams to help you.

- 0.209+0.01
- 10.2+1.1456
- The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
- Select a Block tool, and then click on the screen to place it.
- Click on the Move tool (the arrow) when you are done choosing blocks.
- Subtract by deleting with the delete tool (the trash can), not by crossing out.
Exercise 27.2.3: Subtracting Decimals of Different Lengths
To represent 0.4−0.03, Diego and Noah drew different diagrams. Each rectangle represented 0.1. Each square represented 0.01.
- Diego started by drawing 4 rectangles to represent 0.4. He then replaced 1 rectangle with 10 squares and crossed out 3 squares to represent subtraction of 0.03, leaving 3 rectangles and 7 squares in his diagram.

- Noah started by drawing 4 rectangles to represent 0.4. He then crossed out 3 of rectangles to represent the subtraction, leaving 1 rectangle in his diagram.

- Do you agree that either diagram correctly represents 0.4−0.03? Discuss your reasoning with a partner.
- To represent 0.4−0.03, Elena drew another diagram. She also started by drawing 4 rectangles. She then replaced all 4 rectangles with 40 squares and crossed out 3 squares to represent subtraction of 0.03, leaving 37 squares in her diagram. Is her diagram correct? Discuss your reasoning with a partner.

- Find each difference. If you get stuck, you can use the applet to represent each expression and find its value.
- 0.3−0.05
- 2.1−0.4
- 1.03−0.06
- 0.02−0.007
Be prepared to explain your reasoning.
- The applet has tools that create each of the base-ten blocks. This time you need to decide the value of each block before you begin.
- Select a Block tool, and then click on the screen to place it.
- Click on the Move tool (the arrow) when you are done choosing blocks.
- Subtract by deleting with the delete tool (the trash can), not by crossing out.
Are you ready for more?
A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.
At the Auld Shoppe, a shopper buys items that are worth 2 yellow jewels, 2 green jewels, 2 blue jewels, and 1 indigo jewel. If they came into the store with 1 red jewel, 1 yellow jewel, 2 green jewels, 1 blue jewel, and 2 violet jewels, what jewels do they leave with? Assume the shopkeeper gives them their change using as few jewels as possible.
Summary
Base-ten diagrams can help us understand subtraction as well. Suppose we are finding 0.23−0.07. Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.

Subtracting 7 hundredths means removing 7 small squares, but we do not have enough to remove. Because 1 tenth is equal to 10 hundredths, we can “unbundle” (or decompose) one of the tenths (1 rectangle) into 10 hundredths (10 small squares).

We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.

We have 1 tenth and 6 hundredths remaining, so 0.23−0.07=0.16.

Here is a vertical calculation of 0.23−0.07.

Notice how this representation also shows a tenth is unbundled (or decomposed) into 10 hundredths in order to subtract 7 hundredths.
This works for any decimal place. Suppose we are finding 0.023−0.007. Here is a diagram showing 0.023.

We want to remove 7 thousandths (7 small rectangles). We can “unbundle” (or decompose) one of the hundredths into 10 thousandths.

Now we can remove 7 thousandths.

We have 1 hundredth and 6 thousandths remaining, so 0.023−0.007=0.016.

Here is a vertical calculation of 0.023−0.007.

Practice
Exercise 27.2.4
Here is a base-ten diagram that represents 1.13. Use the diagram to find 1.13−0.46.
Explain or show your reasoning.

Exercise 27.2.5
Compute the following sums. If you get stuck, consider drawing base-ten diagrams.
- 0.027+0.004
- 0.203+0.01
- 1.2+0.145
Exercise 27.2.6
A student said we cannot subtract 1.97 from 20 because 1.97 has two decimal digits and 20 has none. Do you agree with him? Explain or show your reasoning.
Exercise 27.2.7
Decide which calculation shows the correct way to find 0.3−0.006 and explain your reasoning.

Exercise 27.2.8
Complete the calculations so that each shows the correct difference.

Exercise 27.2.9
The school store sells pencils for $0.30 each, hats for $14.50 each, and binders for $3.20 each. Elena would like to buy 3 pencils, a hat, and 2 binders. She estimated that the cost will be less than $20.
- Do you agree with her estimate? Explain your reasoning.
- Estimate the number of pencils could she buy with $5. Explain or show your reasoning.
(From Unit 5.1.1)
Exercise 27.2.10
A rectangular prism measures 712 cm by 12 cm by 1512 cm.
- Calculate the number of cubes with edge length 12 cm that fit in this prism.
- What is the volume of the prism in cm3? Show your reasoning. If you are stuck, think about how many cubes with 12-cm edge lengths fit into 1 cm2.
(From Unit 4.4.4)
Exercise 27.2.11
At a constant speed, a car travels 75 miles in 60 minutes. How far does the car travel in 18 minutes? If you get stuck, consider using the table.
minutes | distance in miles |
---|---|
60 | 75 |
6 | |
18 |
(From Unit 2.4.2)