27.2: Adding and Subtracting Decimals with Few Non-Zero Digits
Lesson
Let's add and subtract decimals.
Exercise \(\PageIndex{1}\): Do the Zeros Matter?
- Evaluate mentally: \(1.009+0.391\)
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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 \(\PageIndex{2}\): 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 \(\PageIndex{3}\): 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.
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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 \(\PageIndex{4}\)
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 \(\PageIndex{5}\)
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 \(\PageIndex{6}\)
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 \(\PageIndex{7}\)
Decide which calculation shows the correct way to find \(0.3-0.006\) and explain your reasoning.
Exercise \(\PageIndex{8}\)
Complete the calculations so that each shows the correct difference.
Exercise \(\PageIndex{9}\)
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 \(\PageIndex{10}\)
A rectangular prism measures \(7\frac{1}{2}\) cm by \(12\) cm by \(15\frac{1}{2}\) cm.
- Calculate the number of cubes with edge length \(\frac{1}{2}\) cm that fit in this prism.
- What is the volume of the prism in \(\text{cm}^{3}\)? Show your reasoning. If you are stuck, think about how many cubes with \(\frac{1}{2}\)-cm edge lengths fit into \(1\text{ cm}^{2}\).
(From Unit 4.4.4)
Exercise \(\PageIndex{11}\)
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)