Here are the step-by-step solutions for each problem on the worksheet.
1)
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Numerator: $\frac{11}{8} \cdot \frac{t-2}{4} = \frac{11(t-2)}{32}$
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Denominator: $\frac{1}{8} - \frac{t-2}{4}$. To subtract, find a common denominator of 8: $\frac{1}{8} - \frac{2(t-2)}{8} = \frac{1 - (2t - 4)}{8} = \frac{1 - 2t + 4}{8} = \frac{5 - 2t}{8}$
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Divide: $\frac{\frac{11(t-2)}{32}}{\frac{5-2t}{8}} = \frac{11(t-2)}{32} \cdot \frac{8}{5-2t}$
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Simplify: The 8 and 32 reduce to 1 and 4.
$$ \frac{11(t-2)}{4(5-2t)} $$
2)
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Numerator: $\frac{1}{z-1} \cdot \frac{5}{z+4} = \frac{5}{(z-1)(z+4)}$
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Denominator: $\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$
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Divide: $\frac{5}{(z-1)(z+4)} \div \frac{1}{3} = \frac{5}{(z-1)(z+4)} \cdot \frac{3}{1}$
$$ \frac{15}{(z-1)(z+4)} $$
3)
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Numerator: $\frac{11}{3} \cdot \frac{6}{c^2}$. Cancel the 3 and 6 (leaving 2 on top): $\frac{11 \cdot 2}{c^2} = \frac{22}{c^2}$
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Denominator: $c^2$
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Divide: $\frac{22}{c^2} \div c^2 = \frac{22}{c^2} \cdot \frac{1}{c^2}$
$$ \frac{22}{c^4} $$
4)
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Numerator: $9$
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Denominator: $\frac{8}{9} + \frac{9}{4}$. Common denominator is 36.
$\frac{8 \cdot 4}{36} + \frac{9 \cdot 9}{36} = \frac{32}{36} + \frac{81}{36} = \frac{113}{36}$
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Divide: $9 \div \frac{113}{36} = 9 \cdot \frac{36}{113}$
$$ \frac{324}{113} $$
5)
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Numerator: $\frac{1}{4} \cdot \frac{d+6}{5} = \frac{d+6}{20}$
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Denominator: $d^2 + \frac{5}{4}$. Common denominator is 4: $\frac{4d^2}{4} + \frac{5}{4} = \frac{4d^2+5}{4}$
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Divide: $\frac{d+6}{20} \div \frac{4d^2+5}{4} = \frac{d+6}{20} \cdot \frac{4}{4d^2+5}$
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Simplify: The 4 and 20 reduce to 1 and 5.
$$ \frac{d+6}{5(4d^2+5)} $$
6)
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Numerator: $\frac{11}{6} + \frac{5}{6} = \frac{16}{6}$
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Denominator: $6$
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Divide: $\frac{16}{6} \div 6 = \frac{16}{6} \cdot \frac{1}{6} = \frac{16}{36}$
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Simplify: Divide top and bottom by 4.
$$ \frac{4}{9} $$
7)
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Numerator: $\frac{3}{2} \cdot \frac{3}{5} = \frac{9}{10}$
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Denominator: $5$
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Divide: $\frac{9}{10} \div 5 = \frac{9}{10} \cdot \frac{1}{5}$
$$ \frac{9}{50} $$
8)
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Numerator: $9$
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Denominator: $\frac{4}{7q} + \frac{7}{q}$. Common denominator is $7q$.
$\frac{4}{7q} + \frac{7 \cdot 7}{7q} = \frac{4 + 49}{7q} = \frac{53}{7q}$
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Divide: $9 \div \frac{53}{7q} = 9 \cdot \frac{7q}{53}$
$$ \frac{63q}{53} $$
9)
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Numerator: $8$
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Denominator: $\frac{11}{8} + \frac{1}{8} = \frac{12}{8}$
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Divide: $8 \div \frac{12}{8} = 8 \cdot \frac{8}{12} = \frac{64}{12}$
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Simplify: Divide top and bottom by 4.
$$ \frac{16}{3} $$
10)
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Numerator: $h^2 + \frac{5}{8}$. Common denominator is 8: $\frac{8h^2}{8} + \frac{5}{8} = \frac{8h^2+5}{8}$
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Denominator: $\frac{11}{8} + \frac{h+3}{9}$. Common denominator is 72.
$\frac{11 \cdot 9}{72} + \frac{8(h+3)}{72} = \frac{99 + 8h + 24}{72} = \frac{8h + 123}{72}$
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Divide: $\frac{8h^2+5}{8} \div \frac{8h+123}{72} = \frac{8h^2+5}{8} \cdot \frac{72}{8h+123}$
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Simplify: The 8 and 72 reduce to 1 and 9.
$$ \frac{9(8h^2+5)}{8h+123} $$
Final Answer:
1) $\frac{11(t-2)}{4(5-2t)}$
2) $\frac{15}{(z-1)(z+4)}$
3) $\frac{22}{c^4}$
4) $\frac{324}{113}$
5) $\frac{d+6}{5(4d^2+5)}$
6) $\frac{4}{9}$
7) $\frac{9}{50}$
8) $\frac{63q}{53}$
9) $\frac{16}{3}$
10) $\frac{9(8h^2+5)}{8h+123}$
Parent Tip: Review the logic above to help your child master the concept of complex fraction worksheet.
