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Answer :
To convert the given decimals to rational numbers, we can express each decimal as a fraction. Here's a step-by-step explanation:
a. 0.823
This is a terminating decimal. To convert it to a fraction, you can consider it as [tex]\(\frac{823}{1000}\)[/tex] since 823 is the number without the decimal point, and it has three decimal places (so the denominator is 1000). Therefore, the rational number is [tex]\(\frac{823}{1000}\)[/tex].
b. 0.275275275...
This is a repeating decimal, with "275" as the repeating part. To express this as a fraction, you can set [tex]\(x = 0.275275275...\)[/tex]. Multiply both sides by 1000 (since "275" has three digits), giving you [tex]\(1000x = 275.275275...\)[/tex]. Subtract the equation [tex]\(x = 0.275275...\)[/tex] from this equation to eliminate the repeating part:
[tex]\[ 1000x - x = 275 \][/tex]
[tex]\[ 999x = 275 \][/tex]
Solve for [tex]\(x\)[/tex], and you'll get:
[tex]\[ x = \frac{275}{999} \][/tex]
c. 0.115
This is another terminating decimal. You can write it as [tex]\(\frac{115}{1000}\)[/tex] because it has three decimal places. Simplifying [tex]\(\frac{115}{1000}\)[/tex] gives [tex]\(\frac{23}{200}\)[/tex].
d. 3.327
This is also a terminating decimal. As a fraction, you can express it as [tex]\(\frac{3327}{1000}\)[/tex] because it has three decimal places. So, the rational number is [tex]\(\frac{3327}{1000}\)[/tex].
e. 0.274
This is a terminating decimal too. Write it as [tex]\(\frac{274}{1000}\)[/tex], since it has three decimal places. Simplifying [tex]\(\frac{274}{1000}\)[/tex] gives [tex]\(\frac{137}{500}\)[/tex].
f. 1.75757575...
This is a repeating decimal, with "75" as the repeating part. Let's set [tex]\(y = 1.75757575...\)[/tex]. Multiply both sides by 100 to get [tex]\(100y = 175.757575...\)[/tex]. Subtract the equation [tex]\(y = 1.757575...\)[/tex] from this equation:
[tex]\[ 100y - y = 175 - 1 \][/tex]
[tex]\[ 99y = 174 \][/tex]
Solve for [tex]\(y\)[/tex], you'll find:
[tex]\[ y = \frac{174}{99} \][/tex]
Simplifying [tex]\(\frac{174}{99}\)[/tex] gives [tex]\(\frac{58}{33}\)[/tex].
In this way, you can convert each decimal into a rational number. Here are the results:
- a. [tex]\(\frac{823}{1000}\)[/tex]
- b. [tex]\(\frac{275}{999}\)[/tex]
- c. [tex]\(\frac{23}{200}\)[/tex]
- d. [tex]\(\frac{3327}{1000}\)[/tex]
- e. [tex]\(\frac{137}{500}\)[/tex]
- f. [tex]\(\frac{58}{33}\)[/tex]
a. 0.823
This is a terminating decimal. To convert it to a fraction, you can consider it as [tex]\(\frac{823}{1000}\)[/tex] since 823 is the number without the decimal point, and it has three decimal places (so the denominator is 1000). Therefore, the rational number is [tex]\(\frac{823}{1000}\)[/tex].
b. 0.275275275...
This is a repeating decimal, with "275" as the repeating part. To express this as a fraction, you can set [tex]\(x = 0.275275275...\)[/tex]. Multiply both sides by 1000 (since "275" has three digits), giving you [tex]\(1000x = 275.275275...\)[/tex]. Subtract the equation [tex]\(x = 0.275275...\)[/tex] from this equation to eliminate the repeating part:
[tex]\[ 1000x - x = 275 \][/tex]
[tex]\[ 999x = 275 \][/tex]
Solve for [tex]\(x\)[/tex], and you'll get:
[tex]\[ x = \frac{275}{999} \][/tex]
c. 0.115
This is another terminating decimal. You can write it as [tex]\(\frac{115}{1000}\)[/tex] because it has three decimal places. Simplifying [tex]\(\frac{115}{1000}\)[/tex] gives [tex]\(\frac{23}{200}\)[/tex].
d. 3.327
This is also a terminating decimal. As a fraction, you can express it as [tex]\(\frac{3327}{1000}\)[/tex] because it has three decimal places. So, the rational number is [tex]\(\frac{3327}{1000}\)[/tex].
e. 0.274
This is a terminating decimal too. Write it as [tex]\(\frac{274}{1000}\)[/tex], since it has three decimal places. Simplifying [tex]\(\frac{274}{1000}\)[/tex] gives [tex]\(\frac{137}{500}\)[/tex].
f. 1.75757575...
This is a repeating decimal, with "75" as the repeating part. Let's set [tex]\(y = 1.75757575...\)[/tex]. Multiply both sides by 100 to get [tex]\(100y = 175.757575...\)[/tex]. Subtract the equation [tex]\(y = 1.757575...\)[/tex] from this equation:
[tex]\[ 100y - y = 175 - 1 \][/tex]
[tex]\[ 99y = 174 \][/tex]
Solve for [tex]\(y\)[/tex], you'll find:
[tex]\[ y = \frac{174}{99} \][/tex]
Simplifying [tex]\(\frac{174}{99}\)[/tex] gives [tex]\(\frac{58}{33}\)[/tex].
In this way, you can convert each decimal into a rational number. Here are the results:
- a. [tex]\(\frac{823}{1000}\)[/tex]
- b. [tex]\(\frac{275}{999}\)[/tex]
- c. [tex]\(\frac{23}{200}\)[/tex]
- d. [tex]\(\frac{3327}{1000}\)[/tex]
- e. [tex]\(\frac{137}{500}\)[/tex]
- f. [tex]\(\frac{58}{33}\)[/tex]
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