Unit 1: Working with Decimals
Topic D: Rounding Numbers
If a pair of jeans cost $49.98, what amount would you say if someone asks what you paid for them? You would probably say, “They cost around $50.”
We often round cents to dollars as we go about our lives. You may already have an idea of how to do this. For example, answer these questions.
- About how much do your groceries cost each month?
- About how much does it cost to fill a small car’s gas tank?
Look at your answers. The amount for groceries may be quite large. When you estimated your answer, how did you round the amount? For example, if your real monthly grocery bill was $481.73 you might have said $482 or perhaps $480. Perhaps you even have estimated to the nearest hundred dollars and said, “About $500 a month for groceries.” All those estimates would be correct.
The amount for a tank of gas is less than a month’s groceries. How did you estimate?
For example, a small car may take $54.72 of gas.
If you estimated to the nearest dollar, you would say, “About $55.”
If you estimated to the nearest ten dollars, you would say, “About $50.”
If you rounded to the nearest dollar you would say, “55 dollars.”
We round a number in different ways depending on several things:
- the size of the number we are rounding
- what we are going to do with the number after we have rounded it off
- our own convenience
Carefully review the place value for whole numbers.
| Thousands | Ones | |||||
|---|---|---|---|---|---|---|
| Hundred thousands | Ten thousands | One thousand | Hundreds | Tens | Ones | Decimal |
Rounding Whole Numbers
We round down if the digit to the right is less than 5. We round up if the digit to the right is 5 or more.
- Rounding numbers gives an approximate amount; it is not an accurate figure.
- Use a different form of the equal sign (≈) which means “approximate equality.”
Review: Rounding to the Nearest Ten
- Underline the tens digit
- Look at the digit in the ones place (to the right). You can put an arrow above it to help you find it later.
- If the ones digit is 5 or more, round up. Write the ones digit as zero and increase the tens digit by one.
- If the ones digit is less than 5, round down. The tens digit does not change and the ones digit is written as a zero.
Example A
[latex]\begin{array}{r}\downarrow\\\underline{2}3\end{array}[/latex]
23 is rounded down to 20. The tens digit stays the same.
23 ≈ 20
Here’s another example:
Example B
[latex]\begin{array}{r}\downarrow\\2\underline{8}7\end{array}[/latex]
287 is rounded up to 290. Tthe tens digit increases by 1.
287 ≈ 290.
Exercise 1
Round each of the following to the nearest ten. Use the “approximate equality” sign ≈.
- 46 ≈ 50
- 111 ≈ 110
- 7 ≈
- 116 ≈
- 71 ≈
- 89 ≈
- 96 ≈
- 4 ≈
- 385 ≈
- 108 ≈
- 73 ≈
- 17 ≈
- 361 ≈
- 8 ≈
- 49 ≈
- 148 ≈
Answers to Exercise 1
- 50
- 110
- 10
- 120
- 70
- 90
- 100
- 0
- 390
- 110
- 70
- 20
- 360
- 10
- 50
- 150
Review: Rounding to the Nearest Hundred
- Underline the hundreds digit.
- Look at the digit in the tens place (to the right). You can put an arrow above it to help you find it later.
- If the tens digit is 5 or more, round up. Write the tens and ones digit as zero and increase the hundreds digit by one.
- If the tens digit is less than 5, round down. The hundreds digit does not change and the tens and ones digit is written as a zero.
Example C
[latex]\begin{array}{c}\downarrow\\ \underline{4}73\end{array}[/latex]
473 is rounded up to 500.
473 ≈ 500
Round down if the tens digit is less than 5 and up if it is 5 or more:
- 728 rounded to the nearest hundred is 700. (The tens digit is 2, which is less than 5, so the hundreds digit stays the same.)
- 758 rounded to the nearest hundred is 800. (The tens digit is 5, which is five or more, so the hundreds digit increases by 1.)
Exercise 2
Round each of the following to the nearest HUNDRED. Use the “approximate equality” sign ≈.
- 330 ≈ 300
- 908
- 2795
- 1260
- 742
- 127
- 302
- 945
- 865
- 275
- 590
- 1240
- 214
- 4450
- 98
- 996
Answers to Exercise 2
- 300
- 900
- 2800
- 1300
- 700
- 100
- 300
- 900
- 900
- 300
- 600
- 1200
- 200
- 4500
- 100
- 1000
Review: Rounding to the Nearest Thousand
- Underline the thousands digit
- Look at the digit in the hundreds place (to the right). You can put an arrow above it to help you find it later.
- If the hundreds digit is 5 or more, round up. Write the hundreds, tens and ones digit as zero and increase the thousands digit by one.
- If the hundreds digit is less than 5, round down. The thousands digit does not change and the hundreds, tens and ones digit is written as a zero.
Example D
[latex]\begin{array}{l}\hspace{0.5em}\downarrow\\ \underline{3}485\end{array}[/latex]
3485 is rounded down to 3000.
3485 ≈ 3000
Round down if the hundreds digit is less than 5 and round up if it is 5 or more:
- 2109 rounded to the nearest thousand is 2000. (The hundreds digit is 1, which is less than 5.)
- 2643 rounded to the nearest thousand is 3000. (The hundreds digit is 6, which is more than 5.)
- 0940 rounded to the nearest thousand is 1000. (The hundreds digit is 9, which is more than 5, so the thousands digit increases from 0 to 1.)
- 0465 rounded to the nearest thousand is 0. (The hundreds digit is 4, which is less than 5, so the thousands digit stays at 0.)
Exercise 3
Round each of the following to the nearest thousand. Use the “approximate equality” sign ≈.
- 1760 ≈ 2000
- 6250
- 850
- 320
- 5544
- 1234
- 492
- 6199
- 9883
- 1045
- 7856
- 500
- 1780
- 495
- 9300
- 700
- 2449
- 5555
- 8914
- 85455
- 6475
Answers to Exercise 3
- 2000
- 6000
- 1000
- 0
- 6000
- 1000
- 0
- 6000
- 10000
- 1000
- 8000
- 1000
- 2000
- 0
- 9000
- 1000
- 2000
- 6000
- 9000
- 85000
- 6000
Rounding Decimals to Whole Numbers
Remember, decimals are part of the whole thing. We can round the decimal to the nearest whole number. Rounding to whole numbers means rounding off to the ones place.
When rounding to the whole number:
- Underline the ones digit.
- Look at the digit in the tenths place (to the right). You can put an arrow above it to help you find it later.
- If the tenths digit is 5 or more, round up. Increase the ones digit by one. Do not write a decimal or any decimal digits.
- If the tenths digit is less than 5, round down. The ones digit does not change. Do not write a decimal or any decimal digits.
Example E
[latex]\begin{array}{l}\hspace{1.3em}\downarrow\\ 3\underline{7}.482\end{array}[/latex]
37.482 rounded to the nearest whole number is 37. (The tenths digit is 4, which is less than 5.)
37.482 ≈ 37
[latex]\begin{array}{l}\hspace{1.3em}\downarrow\\ 3\underline{7}.906\end{array}[/latex]
37.906 rounded to the nearest whole number is 38. (The tenths digit is 9, which is more than 5.)
37. 906 ≈ 38
Example F
- Round to a whole number.
[latex]\begin{array}{l}\hspace{5.58em}\downarrow\\ 42.123\rightarrow4\underline{2}.123\approx42\end{array}[/latex] - Round 960.802 to the nearest whole number.
[latex]\begin{array}{l}\hspace{6.6em}\downarrow\\ 960.802\rightarrow96\underline{0}.802\approx961\end{array}[/latex] - Round 39.5 to the nearest ones.
[latex]\begin{array}{l}\hspace{4.6em}\downarrow\\ 39.5\rightarrow3\underline{9}.5\approx40\end{array}[/latex]
Zeros again –
You know that zeros at the end of a decimal do not change the value of the amount. You can add as many as you like.
But when a decimal has been rounded, drop any zeros after the place where you have rounded.
Instead of 39.52 ≈ 40.0, do 39.52 ≈ 40
Instead of 960.802 ≈ 961.000, do 960.802 ≈ 961
Exercise 4
Round each of the following to the nearest whole number. Use the “approximate equality” sign ≈.
- 11.3 ≈ 11
- 2.679
- 403.8
- 7.6
- 65.91
- 22.2
- 3.76
- 9.2
- 1.7
- 0.6
- 2.63
- 5.09
- 19.8
- 2.1
- 0.7
- 74.2
- 3.61
- 12.3
- 34.5
- 17.82
- 2.45
- 1.792
- 2.01
- 5.55
- 10.3
- 9.9
- 8.15
Answers to Exercise 4
- 11
- 3
- 404
- 8
- 66
- 22
- 4
- 9
- 2
- 1
- 3
- 5
- 20
- 2
- 1
- 74
- 4
- 12
- 35
- 18
- 2
- 2
- 2
- 6
- 10
- 10
- 8
Important Information
If these exercises on rounding are becoming tiresome, please do not despair—there is a purpose. When you do operations (+ − × ÷) with decimals, you will often end up with answers in the ten-thousandths place when you really only need the accuracy of a tenth or a hundredth place decimal. If you do decimal operations on a calculator, you may end up with 6 decimal places (millionths)—not too practical if you are working with money and only want two decimal places! You will know how to round the answer to the decimal place you need for that question or situation.
Rounding Decimals to the Nearest Tenth
- Underline the tenths place digit.
- Look at the digit (to the right) in the hundredths place. You can put an arrow above it to help you find it later.
- If the hundredths digit is less than 5, the tenths digit does not change and the hundredths digit (and all other decimals numbers after the hundredths) are not written at all.
- If the hundredths digit is 5 or more, increase the tenths digit by one and write no more decimals in the hundredths spot or after.
Example G
Round to the nearest tenth.
- [latex]\begin{array}{l}\hspace{6.2em}\downarrow\\13.432\rightarrow13.\underline{4}32\approx13.4\end{array}[/latex]
- [latex]\begin{array}{l}\hspace{6.2em}\downarrow\\13.476\rightarrow13.\underline{4}76\approx13.5\end{array}[/latex]
- [latex]\begin{array}{l}\hspace{5.1em}\downarrow\\0.263\rightarrow0.\underline{2}63\approx0.3\end{array}[/latex]
- [latex]\begin{array}{l}\hspace{7.6em}\downarrow\\234.0399\rightarrow234.\underline{0}399\approx234.0\end{array}[/latex]
Exercise 5
Round each of the following to the nearest tenth.
- 4.23 ≈ 4.2
- 5.18
- 8.54
- 16.09
- 3.52
- 4.14
- 6.24
- 1.76
- 1.74
- 7.19
- 2.15
- 1.44
- 3.172
- 9.99
- 5.09
- 4.111
- 6.046
- 0.71
- 3.63
- 9.45
- 12.36
- 202.305
- 2.66
- 9.492
- 7.388
- 5.249
- 2.45
Answers to Exercise 5
- 4.2
- 5.2
- 8.5
- 16.1
- 3.5
- 4.1
- 6.2
- 1.8
- 1.7
- 7.2
- 2.2
- 1.4
- 3.2
- 10.0
- 5.1
- 4.1
- 6.0
- 0.7
- 3.6
- 9.5
- 12.4
- 202.3
- 2.7
- 9.5
- 7.4
- 5.2
- 2.5
Rounding Decimals to the Nearest Hundredth
Rounding decimals to the nearest hundredth is similar to rounding to the nearest tenth.
- Underline the hundredths place digit.
- Look at the digit (to the right) in the thousandths place. You can put an arrow above it to help you find it later.
- If the thousandths digit is less than 5, the hundredths digit does not change and the thousandths digit (and all other decimals numbers after the hundredths) are not written at all.
- If the thousandths digit is 5 or more, increase the hundredths digit by one and write no more decimals in the thousandths spot or after.
Example H
Round to the nearest hundredth.
[latex]\begin{array}{l}\hspace{7.1em}\downarrow\\35.4524\rightarrow35.4\underline{5}24\approx35.45\end{array}[/latex]
[latex]\begin{array}{l}\hspace{7.1em}\downarrow\\35.4567\rightarrow35.4\underline{5}67\approx35.46\end{array}[/latex]
[latex]\begin{array}{l}\hspace{7.1em}\downarrow\\47.9873\rightarrow47.9\underline{8}73\approx47.99\end{array}[/latex]
[latex]\begin{array}{l}\hspace{7.6em}\downarrow\\23.99609\rightarrow23.9\underline{9}609\approx24.00\end{array}[/latex]
These zeros are significant.
Exercise 6
Round to the nearest hundredth. Keep significant zeros!
- 128.409 ≈ 128.41
- 0.909
- 98.024
- 3.001
- 76.3333
- 0.229
- 100.999
- 0.756
Answers to Exercise 6
- 128.41
- 0.91
- 98.02
- 3.00
- 76.33
- 0. 23
- 101.00
- 0.76
More Dollars and Cents
A cent is what fraction of a dollar?
Yes, a cent is [latex]\tfrac{1}{100}[/latex] of a dollar (one hundredth).
You may be asked to round amounts of money to the nearest cent. What you are actually doing is rounding to the nearest hundredth of a dollar.
- [latex]\begin{array}{l}\hspace{2.3em}\downarrow\\\$3.2\underline{8}6\approx\$3.29\end{array}[/latex]
- [latex]\begin{array}{l}\hspace{2.7em}\downarrow\\\$14.9\underline{2}3\approx\$14.92\end{array}[/latex]
one cent = one hundredth of a dollar
Exercise 7
Round to the nearest cent.
- $42.008 ≈ $42.01
- $ 0.233 ≈ $0.23
- $25.255
- $10.141
- $0.706
- $100.999
- $0.9834
- $2.8977
Answers to Exercise 7
- $42.01
- $0.23
- $25.26
- $10.14
- $0.71
- $101.00
- $0.98
- $2.90
Rounding Decimals to the Nearest Thousandth
Example I
Round to the nearest thousandth (1000th).
2.0486 ⇒ 2.0486 ≈ 2.049
Round to the nearest thousandth (1000th).
29.4324 ⇒ 29.4324 ≈ 29.432
Use rounded numbers to estimate answers in daily situations, in math problem solving, and to get an idea of the answer before you figure something out on a calculator. Numbers that are rounded off make calculations simpler.
Exercise 8
Round the following numbers as called for at the left of the chart.
- Round to the nearest tenth.
- 2.34 ≈ 2.3
- 3.75
- 1.028
- Round to the nearest thousandth.
- 0.1234
- 1.8032
- 7.0052
- Round to the nearest whole number.
- 21.1
- 2.7
- 12.05
- Round to the nearest hundred.
- 275
- 490
- 1260
- Round to the nearest hundredth.
- 1.732
- 2.466
- 3.074
- Round to the nearest ten.
- 68
- 32
- 824
- Round to the nearest thousandth.
- 0.7286
- 0.5027
- 1.2345
Answers to Exercise 8
- Round to the nearest tenth.
- 2.3
- 3.8
- 1.0
- Round to the nearest thousandth.
- 0.123
- 1.803
- 7.005
- Round to the nearest whole number.
- 21
- 3
- 12
- Round to the nearest hundred.
- 300
- 500
- 1300
- Round to the nearest hundredth.
- 1.73
- 2.47
- 3.07
- Round to the nearest ten.
- 70
- 30
- 820
- Round to the nearest thousandth.
- 0.729
- 0.503
- 1.235
Exercise 9
Round the numbers to estimate the answer. Circle the estimate that is the best answer.
- 47 × 52 ≈ 240, 2500 , 250, 2600
Estimation: 50 × 50 = 2500 - 3.2 × 4.875 ≈ 6, 8, 15, 17
- 4149 ÷ 20 ≈ 2000, 200, 20, 230
- 2895 + 2895 ≈ 600, 6000, 4000, 5000
- 91 × 79 ≈ 720, 800, 8000, 80000
- 347 ÷ 50 ≈ 7, 70, 700, 8
- 4892 − 3012 ≈ 1500, 1000, 2000, 2500
- Nathan drives to Terrace and back once a week. He averages 286 km per week. Estimate how many kilometres he drives in one year (52 weeks).
Answers to Exercise 9
- 2500
- 15
- 200
- 6000
- 8000
- 7
- 2000
- 15000 km
Topic D: Self-Test
Mark /10 Aim 8/10
- Round to the nearest hundred. (2 marks)
- 749
- 691
- Round to the nearest whole number. (2 marks)
- 0.831
- 6.24
- Round to the nearest tenth. (2 marks)
- 8.29
- 6.533
- Round to the nearest hundredth. (2 marks)
- 34.792
- 6.459
- Round to the nearest thousandth. (2 marks)
- 5.4392
- 0.8208
- Estimate the answer. (2 marks)
- Mary baby-sat for her twin nephews for 6.75 hours on Saturday. She is paid $8.40 an hour. Estimate her earnings by rounding the numbers in the problem to whole numbers. Show how you worked out the estimate.
Answers to Topic H Self-Test
- Round to the nearest hundred. (2 marks)
- 700
- 700
- Round to the nearest whole number. (2 marks)
- 1
- 6
- Round to the nearest tenth. (2 marks)
- 8.3
- 6.5
- Round to the nearest hundredth. (2 marks)
- 34.79
- 6.46
- Round to the nearest thousandth. (2 marks)
- 5.439
- 0.821
- Estimate the answer. (2 marks)
- Estimation: 7 hours × $8 ≈ $56
We understand numbers by the way the digits (numerals) are arranged in relationship to each other and to the decimal point. Each position has a certain value. Our number system is a decimal system. The place value is based on ten.
Any of the ten numerals (0 to 9) are digits. This term comes from our ten fingers which are called digits. The numerals came to be called "digits" from the practice of counting on the fingers!