Testinar Grade 8 math

Turning Repeating Decimals into Fractions

Learn to reason about and solve Grade 8 problems involving Turning Repeating Decimals into Fractions.

CH01-02 Number Systems and Financial Reasoning

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Which multiplication step is shown here?
1. Multiply by 10 to shift 1 decimal place.
2. Multiply by 26 to isolate the repeating part.
3. Multiply by 1000 to shift 3 decimal places.
4. Multiply by 100 to shift 2 decimal places.
Based on the diagram, what is the denominator when \(0.\overline{142}\) is converted?
1. \(99\)
2. \(142\)
3. \(999\)
4. \(1000\)
Using the diagram, what is \(100x - 10x\)?
1. \(34.\overline{0}\)
2. \(37\)
3. \(33\)
4. \(34\)
What is \(0.2\overline{5}\) as a fraction?
1. \(\frac{25}{100}\)
2. \(\frac{25}{90}\)
3. \(\frac{2}{10}\)
4. \(\frac{23}{90}\)
What goes in the numerator when converting \(0.7\overline{89}\)?
1. \(7 + 89 = 96\)
2. \(789\)
3. \(89\)
4. \(789 - 7 = 782\)
Write \(0.\overline{4}\) as a fraction in simplest form.
1. \(\frac{2}{5}\)
2. \(\frac{4}{10}\)
3. \(\frac{4}{99}\)
4. \(\frac{4}{9}\)
Convert \(0.\overline{7}\) to a fraction.
1. \(\frac{1}{9}\)
2. \(\frac{7}{10}\)
3. \(\frac{7}{99}\)
4. \(\frac{7}{9}\)
What is \(0.\overline{3}\) as a fraction?
1. \(\frac{1}{10}\)
2. \(\frac{3}{10}\)
3. \(\frac{3}{99}\)
4. \(\frac{3}{9}=\frac{1}{3}\)

Teacher and parent reference

Answer key and explanations

D. Multiply by 100 to shift 2 decimal places.
The repeating block \(26\) has 2 digits, so we multiply by \(10^2=100\). This shifts the decimal 2 places to align the repeating parts for subtraction.
C. \(999\)
A 3-digit repeating block always produces a denominator of \(999 = 10^3 - 1\).
D. \(34\)
\(100x - 10x = 37.\overline{7} - 3.\overline{7} = 34\). The repeating parts cancel, leaving \(90x = 34\).
D. \(\frac{23}{90}\)
Following the algorithm: \(100x - 10x = 25.\overline{5} - 2.\overline{5} = 23\), so \(90x = 23\) and \(x = \frac{23}{90}\).
D. \(789 - 7 = 782\)
Let \(x=0.7\overline{89}\). Then \(10x=7.\overline{89}\) and \(1000x=789.\overline{89}\). Subtracting gives \(990x=789-7=782\), so the numerator is \(782\).
D. \(\frac{4}{9}\)
Let \(x=0.\overline{4}\). Then \(10x=4.\overline{4}\). Subtracting: \(10x-x=4.\overline{4}-0.\overline{4}=4\), so \(9x=4\) and \(x=\frac{4}{9}\).
D. \(\frac{7}{9}\)
Let \(x=0.\overline{7}\). Multiply by 10: \(10x=7.\overline{7}\). Subtract: \(10x-x=7\), so \(9x=7\) and \(x=\frac{7}{9}\).
D. \(\frac{3}{9}=\frac{1}{3}\)
Let \(x=0.\overline{3}\). Then \(10x=3.\overline{3}\). Subtracting: \(9x=3\), so \(x=\frac{3}{9}=\frac{1}{3}\).