Question

dividing rational expressions
simplify each expression.

1. $\frac{10r}{8} div \frac{3}{2}$
2. $\frac{11d}{d-2} div \frac{11d}{5d-10}$
3. $\frac{2c}{c+8} div \frac{2c}{10c+80}$
4. $\frac{18s-36}{10} div \frac{24s-48}{10s}$
5. $\frac{132z-77}{5} div \frac{96z-56}{5z}$
6. $\frac{12h^2}{4} div \frac{10h}{7}$
7. $\frac{14y-10}{4} div \frac{21y-15}{4y}$
8. $\frac{7q}{q+11} div \frac{7q}{9q+99}$
9. $\frac{3b}{b-5} div \frac{3b}{12b-60}$
10. $\frac{4n}{8} div \frac{9}{12}$

Explanation:

Step1: Rewrite division as multiplication

To divide rational expressions, multiply by the reciprocal of the divisor.

For 1) $\frac{10r}{8} \div \frac{3}{2}$

Expression: $\frac{10r}{8} \times \frac{2}{3}$

Step2: Simplify and multiply

Cancel common factors, then multiply numerators and denominators.
Expression: $\frac{10r \times 2}{8 \times 3} = \frac{20r}{24} = \frac{5r}{6}$

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Step1: Rewrite division as multiplication

Multiply by the reciprocal of the divisor, factor denominator first.

For 2) $\frac{11d}{d-2} \div \frac{11d}{5d-10}$

Expression: $\frac{11d}{d-2} \times \frac{5(d-2)}{11d}$

Step2: Cancel common factors

Eliminate matching terms in numerator and denominator.
Expression: $\frac{11d \times 5(d-2)}{(d-2) \times 11d} = 5$

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Step1: Rewrite division as multiplication

Factor the second denominator, then multiply by reciprocal.

For 3) $\frac{2c}{c+8} \div \frac{2c}{10c+80}$

Expression: $\frac{2c}{c+8} \times \frac{10(c+8)}{2c}$

Step2: Cancel common factors

Eliminate matching terms in numerator and denominator.
Expression: $\frac{2c \times 10(c+8)}{(c+8) \times 2c} = 10$

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Step1: Rewrite division as multiplication

Factor numerators, then multiply by reciprocal of divisor.

For 4) $\frac{18s-36}{10} \div \frac{24s-48}{10s}$

Expression: $\frac{18(s-2)}{10} \times \frac{10s}{24(s-2)}$

Step2: Cancel common factors

Eliminate matching terms, simplify remaining fractions.
Expression: $\frac{18(s-2) \times 10s}{10 \times 24(s-2)} = \frac{18s}{24} = \frac{3s}{4}$

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Step1: Rewrite division as multiplication

Factor numerators, then multiply by reciprocal of divisor.

For 5) $\frac{132z-77}{5} \div \frac{96z-56}{5z}$

Expression: $\frac{11(12z-7)}{5} \times \frac{5z}{8(12z-7)}$

Step2: Cancel common factors

Eliminate matching terms, simplify remaining terms.
Expression: $\frac{11(12z-7) \times 5z}{5 \times 8(12z-7)} = \frac{11z}{8}$

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Step1: Rewrite division as multiplication

Multiply by the reciprocal of the divisor.

For 6) $\frac{12h^2}{4} \div \frac{10h}{7}$

Expression: $\frac{12h^2}{4} \times \frac{7}{10h}$

Step2: Simplify and multiply

Cancel common factors, then multiply numerators and denominators.
Expression: $\frac{12h^2 \times 7}{4 \times 10h} = \frac{84h^2}{40h} = \frac{21h}{10}$

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Step1: Rewrite division as multiplication

Factor numerators, then multiply by reciprocal of divisor.

For 7) $\frac{14y-10}{4} \div \frac{21y-15}{4y}$

Expression: $\frac{2(7y-5)}{4} \times \frac{4y}{3(7y-5)}$

Step2: Cancel common factors

Eliminate matching terms, simplify remaining terms.
Expression: $\frac{2(7y-5) \times 4y}{4 \times 3(7y-5)} = \frac{8y}{12} = \frac{2y}{3}$

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Step1: Rewrite division as multiplication

Factor the second denominator, then multiply by reciprocal.

For 8) $\frac{7q}{q+11} \div \frac{7q}{9q+99}$

Expression: $\frac{7q}{q+11} \times \frac{9(q+11)}{7q}$

Step2: Cancel common factors

Eliminate matching terms in numerator and denominator.
Expression: $\frac{7q \times 9(q+11)}{(q+11) \times 7q} = 9$

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Step1: Rewrite division as multiplication

Factor the second denominator, then multiply by reciprocal.

For 9) $\frac{3b}{b-5} \div \frac{3b}{12b-60}$

Expression: $\frac{3b}{b-5} \times \frac{12(b-5)}{3b}$

Step2: Cancel common factors

Eliminate matching terms in numerator and denominator.
Expression: $\frac{3b \times 12(b-5)}{(b-5) \times 3b} = 12$

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Step1: Rewrite division as multiplication

Multiply by the reciprocal of the divisor, simplify fractions first.

For 10) $\frac{4n}{8} \div \frac{9}{12}$

Expression: $\frac{n}{2} \times \frac{4}{3}$

Step2: Multiply and simplify

Multiply numerators and denominators, reduce the fraction.
Expression: $\frac{n \times 4}{2 \times 3} = \frac{4n}{6} = \frac{2n}{3}$

Answer:

1. $\frac{5r}{6}$
2. $5$
3. $10$
4. $\frac{3s}{4}$
5. $\frac{11z}{8}$
6. $\frac{21h}{10}$
7. $\frac{2y}{3}$
8. $9$
9. $12$
10. $\frac{2n}{3}$