Fraction and Decimal worksheet

1. Exact decimal: 0.(384615)

1. The denominator has a factor (13) other than 2 and 5, so the decimal repeats forever

   $13$$=\text{13}$

2. Divide the numerator by the denominator to get the whole-number part and a remainder

   $5$$=0\times 13+5$

3. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 3 is the next digit

   $10\times 5$$=3\times 13+11$

4. Decimal place 2: multiply the remainder by 10 and divide again; the quotient 8 is the next digit

   $10\times 11$$=8\times 13+6$

5. Decimal place 3: multiply the remainder by 10 and divide again; the quotient 4 is the next digit

   $10\times 6$$=4\times 13+8$

6. Decimal place 4: multiply the remainder by 10 and divide again; the quotient 6 is the next digit

   $10\times 8$$=6\times 13+2$

7. Decimal place 5: multiply the remainder by 10 and divide again; the quotient 1 is the next digit

   $10\times 2$$=1\times 13+7$

8. Decimal place 6: multiply the remainder by 10 and divide again; the quotient 5 is the next digit

   $10\times 7$$=5\times 13+5$

9. The remainder 5 has appeared before (at decimal place 1), so the 6 digits from there repeat forever; the repeating block is written in parentheses

   $\frac{5}{13}$$=\text{0.(384615)}$

2. Exact decimal: 0.4

1. Reduce the fraction first; it has the same decimal and a smaller denominator

   $\frac{4}{10}$$=\frac{2}{5}$

2. The denominator's only prime factors are 2 and 5, so the decimal terminates

   $5$$=\text{5^1}$

3. Divide the numerator by the denominator to get the whole-number part and a remainder

   $2$$=0\times 5+2$

4. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 4 is the next digit

   $10\times 2$$=4\times 5+0$

5. The remainder reached 0, so the decimal ends here

   $\frac{2}{5}$$=\text{0.4}$

3. Fraction (lowest terms): 69/200

1. There are 3 digits after the decimal point, so write the digits over 10 to the power 3 = 1,000

   $\text{0.345}$$=\frac{345}{\mathrm{1,000}}$

2. Find the biggest number that divides both the top and bottom evenly, called the GCD. Divide, keep what's left over, and repeat until nothing is left

   $\text{GCD(345, 1,000)}$

3. 1,000 ÷ 345 leaves 310 left over

   $\mathrm{1,000}$$=2\times 345+310$

4. 345 ÷ 310 leaves 35 left over

   $345$$=1\times 310+35$

5. 310 ÷ 35 leaves 30 left over

   $310$$=8\times 35+30$

6. 35 ÷ 30 leaves 5 left over

   $35$$=1\times 30+5$

7. Nothing is left over, so the last number you divided by, 5, is the GCD

   $30$$=6\times 5+0$

8. Divide the top and bottom by the GCD, 5, to get the simplest form

   $\frac{345÷5}{\mathrm{1,000}÷5}$$=\frac{69}{200}$

9. That's the decimal as a fraction in lowest terms

   $\text{0.345}$$=\frac{69}{200}$

4. Fraction (lowest terms): 2/5

1. There are 3 digits after the decimal point, so write the digits over 10 to the power 3 = 1,000

   $\text{0.400}$$=\frac{400}{\mathrm{1,000}}$

2. Find the biggest number that divides both the top and bottom evenly, called the GCD. Divide, keep what's left over, and repeat until nothing is left

   $\text{GCD(400, 1,000)}$

3. 1,000 ÷ 400 leaves 200 left over

   $\mathrm{1,000}$$=2\times 400+200$

4. Nothing is left over, so the last number you divided by, 200, is the GCD

   $400$$=2\times 200+0$

5. Divide the top and bottom by the GCD, 200, to get the simplest form

   $\frac{400÷200}{\mathrm{1,000}÷200}$$=\frac{2}{5}$

6. That's the decimal as a fraction in lowest terms

   $\text{0.400}$$=\frac{2}{5}$

5. Exact decimal: 0.(3)

1. Reduce the fraction first; it has the same decimal and a smaller denominator

   $\frac{5}{15}$$=\frac{1}{3}$

2. The denominator has a factor (3) other than 2 and 5, so the decimal repeats forever

   $3$$=\text{3}$

3. Divide the numerator by the denominator to get the whole-number part and a remainder

   $1$$=0\times 3+1$

4. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 3 is the next digit

   $10\times 1$$=3\times 3+1$

5. The remainder 1 has appeared before (at decimal place 1), so the digit from there repeat forever; the repeating block is written in parentheses

   $\frac{1}{3}$$=\text{0.(3)}$

6. Exact decimal: 0.(3)

1. Reduce the fraction first; it has the same decimal and a smaller denominator

   $\frac{4}{12}$$=\frac{1}{3}$

2. The denominator has a factor (3) other than 2 and 5, so the decimal repeats forever

   $3$$=\text{3}$

3. Divide the numerator by the denominator to get the whole-number part and a remainder

   $1$$=0\times 3+1$

4. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 3 is the next digit

   $10\times 1$$=3\times 3+1$

5. The remainder 1 has appeared before (at decimal place 1), so the digit from there repeat forever; the repeating block is written in parentheses

   $\frac{1}{3}$$=\text{0.(3)}$

7. Fraction (lowest terms): 19/50

1. There are 3 digits after the decimal point, so write the digits over 10 to the power 3 = 1,000

   $\text{0.380}$$=\frac{380}{\mathrm{1,000}}$

2. Find the biggest number that divides both the top and bottom evenly, called the GCD. Divide, keep what's left over, and repeat until nothing is left

   $\text{GCD(380, 1,000)}$

3. 1,000 ÷ 380 leaves 240 left over

   $\mathrm{1,000}$$=2\times 380+240$

4. 380 ÷ 240 leaves 140 left over

   $380$$=1\times 240+140$

5. 240 ÷ 140 leaves 100 left over

   $240$$=1\times 140+100$

6. 140 ÷ 100 leaves 40 left over

   $140$$=1\times 100+40$

7. 100 ÷ 40 leaves 20 left over

   $100$$=2\times 40+20$

8. Nothing is left over, so the last number you divided by, 20, is the GCD

   $40$$=2\times 20+0$

9. Divide the top and bottom by the GCD, 20, to get the simplest form

   $\frac{380÷20}{\mathrm{1,000}÷20}$$=\frac{19}{50}$

10. That's the decimal as a fraction in lowest terms

    $\text{0.380}$$=\frac{19}{50}$

8. Exact decimal: 0.2(7)

1. The denominator has a factor (9) other than 2 and 5, so the decimal repeats forever

   $18$$=\text{2^1 × 9}$

2. Divide the numerator by the denominator to get the whole-number part and a remainder

   $5$$=0\times 18+5$

3. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 2 is the next digit

   $10\times 5$$=2\times 18+14$

4. Decimal place 2: multiply the remainder by 10 and divide again; the quotient 7 is the next digit

   $10\times 14$$=7\times 18+14$

5. The remainder 14 has appeared before (at decimal place 2), so the digit from there repeat forever; the repeating block is written in parentheses

   $\frac{5}{18}$$=\text{0.2(7)}$

9. Exact decimal: 0.(307692)

1. The denominator has a factor (13) other than 2 and 5, so the decimal repeats forever

   $13$$=\text{13}$

2. Divide the numerator by the denominator to get the whole-number part and a remainder

   $4$$=0\times 13+4$

3. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 3 is the next digit

   $10\times 4$$=3\times 13+1$

4. Decimal place 2: multiply the remainder by 10 and divide again; the quotient 0 is the next digit

   $10\times 1$$=0\times 13+10$

5. Decimal place 3: multiply the remainder by 10 and divide again; the quotient 7 is the next digit

   $10\times 10$$=7\times 13+9$

6. Decimal place 4: multiply the remainder by 10 and divide again; the quotient 6 is the next digit

   $10\times 9$$=6\times 13+12$

7. Decimal place 5: multiply the remainder by 10 and divide again; the quotient 9 is the next digit

   $10\times 12$$=9\times 13+3$

8. Decimal place 6: multiply the remainder by 10 and divide again; the quotient 2 is the next digit

   $10\times 3$$=2\times 13+4$

9. The remainder 4 has appeared before (at decimal place 1), so the 6 digits from there repeat forever; the repeating block is written in parentheses

   $\frac{4}{13}$$=\text{0.(307692)}$

10. Exact decimal: 0.5

1. Reduce the fraction first; it has the same decimal and a smaller denominator

   $\frac{6}{12}$$=\frac{1}{2}$

2. The denominator's only prime factors are 2 and 5, so the decimal terminates

   $2$$=\text{2^1}$

3. Divide the numerator by the denominator to get the whole-number part and a remainder

   $1$$=0\times 2+1$

4. Decimal place 1: multiply the remainder by 10 and divide again; the quotient 5 is the next digit

   $10\times 1$$=5\times 2+0$

5. The remainder reached 0, so the decimal ends here

   $\frac{1}{2}$$=\text{0.5}$