Mental Math to Multiply by Powers of Ten Jen Kershaw Catherine Kwok
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AUTHORS Jen Kershaw Catherine Kwok
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C HAPTER
Chapter 1. Mental Math to Multiply by Powers of Ten
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Mental Math to Multiply by Powers of Ten
In this concept, you will learn to multiply decimals by powers of ten using mental math.
According to the United States Mint, a quarter weighs 5.67 grams. Jeremy has a kitchen scale that can measure up to 5,050 grams. Susan bets Jeremy that they could probably weigh about 1,000 quarters on the kitchen scale. Jeremy thinks that a thousand quarters would be too much for the scale. Who is correct? In this concept, you will learn to multiply decimals by powers of ten using mental math. Powers of Ten
You have already learned to multiply decimal by whole numbers, however, there is a pattern that you can follow when you multiply decimals by powers of ten. Powers of ten are numbers where 10 is multiplied by itself a certain number of times. It can be written in exponent form where the base is 10 and the exponent, n, is the number of times 10 is multiplied by itself.
10n Here are some examples of power of ten numbers. 101 102 103 104
= = = =
1 × 10 = 10 1 × 10 × 10 = 100 1 × 10 × 10 × 10 = 1, 000 1 × 10 × 10 × 10 × 10 = 10, 000
Notice that the exponent is the same as the number of zeros in the number. Let’s look at a decimal number multiplied by power of ten numbers. 3.4 × 10 = 3− 4 → 3.4 × 100 = 340 − → 3.4 × 1000 = 3, 400 −→ 1
www.ck12.org The decimal point is moved to the right once for every 10 multiplied or the number of zeros in the power of ten number. In the first example, 10 has one zero and the decimal point is moved one place to the right. In the second example, 100 has two zeros and the decimal point is moved two places to the right. The third example has three zeros and the decimal point is moved three places to the right. Powers of ten also include decimal values. These powers of ten are also called negative powers of ten. It is written like a power of ten except that the number in the exponent has a negative sign (-). The negative sign tells you to divide by 10 a total of n number of times. Here are some decimal values for negative powers of ten. 10−1 10−2 10−3 10−4
= = = =
1 10
= 0.1
1 10×10 = 0.01 1 10×10×10 = 0.001 1 10×10×10×10 = 0.0001
Let’s look at a decimal number multiplied by decimal power of ten numbers. 3.4 × 0.1 = 0. 3 4 ← − 3.4 × 0.01 = 0.034 ← − 3.4 × 0.001 = 0.0034 ←− The decimal point in the product is moved to the left for every decimal place in the power of ten decimal number. In the first example, there is one decimal place in 0.1 and the decimal point is moved one place to the left. In the second example, there are two decimal places in 0.01 and the decimal point is moved two places to the left. The third example has three decimal places in 0.001 and the decimal point is moved three places to the left. Depending on what type of power of ten is being multiplied, you can calculate the product using mental math. • When multiplying by whole number powers of ten, the number will become larger. Move the decimal to the right once for every zero in the power of ten. • When multiplying by decimal powers of ten, the number will become smaller. Move the decimal to the left once for every decimal place value. Examples Example 1
Earlier, you were given a problem about Jeremy and Susan and their bet. Susan and Jeremy bet on if the kitchen scale can weigh 1,000 quarters. To find the weight of 1,000 quarters, multiply the weight of a quarter by 1,000. A quarter weighs 5.67 grams. First, write an equation to find the weight of 1,000 quarters.
5.67 × 1, 000 = Then, use mental math to find the product. Count the number of zeros in the power of ten number. There are three. Next, move the decimal point three places to the right.
5.67 → 5, 670. 2
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Chapter 1. Mental Math to Multiply by Powers of Ten
Finally, compare the weight of 1,000 quarter to the maximum weight the scale can measure. The kitchen scale can measure up to 5,050 grams.
Quarters Scale 5, 670 g > 5, 050 g The scale will not be able to weigh the quarters. Jeremy is correct. Example 2
Use mental math to find the product.
4.567 × 1000 = First, count the number of zeros in the power of ten number. There are three. Then, move the decimal point three places to the right.
4.567 → 4, 567. The product of 4.567 times 1000 is 4,567.
6.7 × 0.1 = First, count the number of decimal place values in the power of ten number. There is one. Then, move the decimal point once to the left.
6.7 → 0.67 The product of 6.7 and 0.1 is 0.67. Use mental math to find the product for the following problems. Example 3
0.23 × 100 = First, count the number of zeros in the power of ten number. There are two. Then, move the decimal point two places to the right.
0.23 → 23. The product of 0.23 and 100 is 23. 3
www.ck12.org Example 4
213.5 × 0.01 = First, count the number of decimal place values in the power of ten number. There are two. Then, move the decimal point two places to the left.
213.5 → 2.135 The product of 213.5 times 0.01 is 2.135. Example 5
127.3 × 1, 000 = First, count the number of zeros in the power of ten number. There are three. Then, move the decimal point three places to the right.
127.3 → 127, 300. The product of 127.3 times 1,000 is 127,300. Review
Use mental math to find the product for the following problems. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
3.4 × 10 = 0.56 × 10 = 1.234 × 1000 = 87.9 × 100 = 98.32 × 10 = 7.2 × 1000 = 12.5 × 10 = 18.91 × 10 = 5 × 0.01 = 7.7 × 0.001 = 12.4 × 0.1 = 456.8 × 0.001 = 0.8 × 0.01 = 31.5 × 0.0001 = 4.56 × 0.1 = 8.68 × 0.01 =
Answers for Review Problems
To see the Review answers, open this PDF file and look for section 4.13. 4
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Chapter 1. Mental Math to Multiply by Powers of Ten
Resources
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