Question

multiplicative inverses (reciprocals): used to solve multiplication/division equations that contain fractions!
find the multiplicative inverse, or reciprocal of: $\frac{1}{3}$ $\frac{5}{7}$ $\frac{2}{5}$
now let’s use multiplicative inverses to solve equations...
solve $\frac{3}{5}t = 6$
the coefficient of t is $\frac{3}{5}$. the reciprocal of $\frac{3}{5}$ is _____.
$square$ $\frac{3}{5}t = 6$ $square$ multiply each side by the multiplicative inverse.
$t = $_____. simplify.
$t = $_____. solve.
solve $\frac{2}{7}t = 8$
let’s practice!!
solve each equation. check your solution.
$\frac{1}{7}t = 3$ $\frac{4}{5}t = 8$
$\frac{1}{9}t = 6$ $\frac{3}{5}t = 6$
$\frac{2}{3} = \frac{3}{10}t$ $\frac{1}{4}a = \frac{4}{15}$
$\frac{a}{9} = 11$ $\frac{h}{8} = 6$

Explanation:

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Part 1: Find Multiplicative Inverses

Step1: Flip $\frac{1}{3}$ to get reciprocal

$\text{Reciprocal of } \frac{1}{3} = 3$

Step2: Flip $\frac{5}{7}$ to get reciprocal

$\text{Reciprocal of } \frac{5}{7} = \frac{7}{5}$

Step3: Flip $\frac{2}{5}$ to get reciprocal

$\text{Reciprocal of } \frac{2}{5} = \frac{5}{2}$

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Part 2: Solve $\boldsymbol{\frac{3}{5}t=6}$

Step1: Identify reciprocal of $\frac{3}{5}$

$\text{Reciprocal of } \frac{3}{5} = \frac{5}{3}$

Step2: Multiply both sides by $\frac{5}{3}$

$\frac{5}{3} \times \frac{3}{5}t = 6 \times \frac{5}{3}$

Step3: Simplify left side

$1 \times t = 6 \times \frac{5}{3}$

Step4: Calculate right side

$t = 10$

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Part 3: Solve $\boldsymbol{\frac{2}{7}t=8}$

Step1: Identify reciprocal of $\frac{2}{7}$

$\text{Reciprocal of } \frac{2}{7} = \frac{7}{2}$

Step2: Multiply both sides by $\frac{7}{2}$

$\frac{7}{2} \times \frac{2}{7}t = 8 \times \frac{7}{2}$

Step3: Simplify left side

$1 \times t = 8 \times \frac{7}{2}$

Step4: Calculate right side

$t = 28$

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Part 4: Practice Problems

1. Solve $\boldsymbol{\frac{1}{7}t=3}$

Step1: Multiply by reciprocal of $\frac{1}{7}$

$7 \times \frac{1}{7}t = 3 \times 7$

Step2: Simplify to find $t$

$t = 21$

Step3: Check solution

$\frac{1}{7} \times 21 = 3$, which is correct.

2. Solve $\boldsymbol{\frac{4}{5}t=8}$

Step1: Multiply by reciprocal of $\frac{4}{5}$

$\frac{5}{4} \times \frac{4}{5}t = 8 \times \frac{5}{4}$

Step2: Simplify to find $t$

$t = 10$

Step3: Check solution

$\frac{4}{5} \times 10 = 8$, which is correct.

3. Solve $\boldsymbol{\frac{1}{9}t=6}$

Step1: Multiply by reciprocal of $\frac{1}{9}$

$9 \times \frac{1}{9}t = 6 \times 9$

Step2: Simplify to find $t$

$t = 54$

Step3: Check solution

$\frac{1}{9} \times 54 = 6$, which is correct.

4. Solve $\boldsymbol{\frac{3}{5}t=6}$

Step1: Multiply by reciprocal of $\frac{3}{5}$

$\frac{5}{3} \times \frac{3}{5}t = 6 \times \frac{5}{3}$

Step2: Simplify to find $t$

$t = 10$

Step3: Check solution

$\frac{3}{5} \times 10 = 6$, which is correct.

5. Solve $\boldsymbol{\frac{2}{3}=\frac{3}{10}t}$

Step1: Multiply by reciprocal of $\frac{3}{10}$

$\frac{10}{3} \times \frac{2}{3} = \frac{10}{3} \times \frac{3}{10}t$

Step2: Simplify to find $t$

$t = \frac{20}{9}$

Step3: Check solution

$\frac{3}{10} \times \frac{20}{9} = \frac{2}{3}$, which is correct.

6. Solve $\boldsymbol{\frac{1}{4}a=\frac{4}{15}}$

Step1: Multiply by reciprocal of $\frac{1}{4}$

$4 \times \frac{1}{4}a = \frac{4}{15} \times 4$

Step2: Simplify to find $a$

$a = \frac{16}{15}$

Step3: Check solution

$\frac{1}{4} \times \frac{16}{15} = \frac{4}{15}$, which is correct.

7. Solve $\boldsymbol{\frac{a}{9}=11}$

Step1: Rewrite as $\frac{1}{9}a=11$, multiply by 9

$9 \times \frac{1}{9}a = 11 \times 9$

Step2: Simplify to find $a$

$a = 99$

Step3: Check solution

$\frac{99}{9} = 11$, which is correct.

8. Solve $\boldsymbol{\frac{h}{8}=6}$

Step1: Rewrite as $\frac{1}{8}h=6$, multiply by 8

$8 \times \frac{1}{8}h = 6 \times 8$

Step2: Simplify to find $h$

$h = 48$

Step3: Check solution

$\frac{48}{8} = 6$, which is correct.

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Answer:

Multiplicative Inverses:

• $\frac{1}{3}$: $3$
• $\frac{5}{7}$: $\frac{7}{5}$
• $\frac{2}{5}$: $\frac{5}{2}$

Solve $\frac{3}{5}t=6$:

• Reciprocal of $\frac{3}{5}$: $\frac{5}{3}$
• Equation after multiplying: $\boldsymbol{\frac{5}{3}} \frac{3}{5}t = 6 \boldsymbol{\frac{5}{3}}$
• Simplified: $t = \frac{30}{3}$
• Solution: $t=10$

Solve $\frac{2}{7}t=8$:

• Solution: $t=28$

Practice Problems:

1. $\frac{1}{7}t=3$: $t=21$
2. $\frac{4}{5}t=8$: $t=10$
3. $\frac{1}{9}t=6$: $t=54$
4. $\frac{3}{5}t=6$: $t=10$
5. $\frac{2}{3}=\frac{3}{10}t$: $t=\frac{20}{9}$
6. $\frac{1}{4}a=\frac{4}{15}$: $a=\frac{16}{15}$
7. $\frac{a}{9}=11$: $a=99$
8. $\frac{h}{8}=6$: $h=48$