Chapter 3 Addition and Subtraction 2 5.a. For each of the following decimals, show how to write the decimal as a fraction by first put- ting the decimal in expanded form: i. 2.34 ii. 124.5 iii. 7.938 b. Based on your results in part (a), describe a quick way to rewrite a finite decimal as a frac- tion. Illustrate with the example 2748.963
The Correct Answer and Explanation is:
Writing the Decimals as Fractions in Expanded Form:
i. 2.34
- Break the decimal down into expanded form: 2.34=2+310+41002.34 = 2 + \frac{3}{10} + \frac{4}{100}2.34=2+103+1004
- Now, combine the terms into a single fraction: 2.34=2341002.34 = \frac{234}{100}2.34=100234 The fraction is 234100\frac{234}{100}100234, which can be simplified by dividing both the numerator and the denominator by 2: 234100=11750\frac{234}{100} = \frac{117}{50}100234=50117
ii. 124.5
- Break the decimal down into expanded form: 124.5=124+510124.5 = 124 + \frac{5}{10}124.5=124+105
- Now, combine the terms into a single fraction: 124.5=124510124.5 = \frac{1245}{10}124.5=101245 This fraction is already in its simplest form.
iii. 7.938
- Break the decimal down into expanded form: 7.938=7+910+3100+810007.938 = 7 + \frac{9}{10} + \frac{3}{100} + \frac{8}{1000}7.938=7+109+1003+10008
- Now, combine the terms into a single fraction: 7.938=793810007.938 = \frac{7938}{1000}7.938=10007938 This fraction is already in its simplest form, as the greatest common divisor (GCD) of 7938 and 1000 is 2: 79381000=3969500\frac{7938}{1000} = \frac{3969}{500}10007938=5003969
(b) Quick Way to Rewrite a Finite Decimal as a Fraction:
A quick way to rewrite a finite decimal as a fraction is to:
- Remove the decimal point by multiplying both the numerator and denominator by a power of 10 such that the decimal point moves to the rightmost position. The number of places you move the decimal point determines the power of 10.
- Simplify the resulting fraction if possible.
Example: 2748.963
- Rewrite the decimal in expanded form: 2748.963=2748+910+6100+310002748.963 = 2748 + \frac{9}{10} + \frac{6}{100} + \frac{3}{1000}2748.963=2748+109+1006+10003
- Combine the terms: 2748.963=274896310002748.963 = \frac{2748963}{1000}2748.963=10002748963
- This fraction is already in its simplest form as 2748963 and 1000 do not have a common factor.
Thus, the decimal 2748.963 is written as the fraction 27489631000\frac{2748963}{1000}10002748963.
In general, to convert a decimal to a fraction:
- Count the number of digits after the decimal point.
- Multiply the decimal by 10n10^n10n (where nnn is the number of digits after the decimal point) to shift the decimal place.
- Write the resulting number over the corresponding power of 10 and simplify if necessary.
