Question

lesson 4.5 finding common denominators
it is easier to work with fractions if they have common denominators.
$\frac{1}{5}$ and $\frac{3}{5}$ have like, or common, denominators.
$\frac{1}{4}$ and $\frac{3}{5}$ have unlike denominators.
rename these fractions so that they have common denominators by finding the least common multiple (lcm) of their denominators.
4: 4, 8, 12, 16, 20, 24, … 20 is the smallest multiple they have in common.
5: 5, 10, 15, 20, 25, … so, 20 is the lcm.
rename each fraction with a denominator of 20.
$\frac{1}{4} = \frac{}{20}$ $\frac{1}{4} = \frac{1×5}{4×5} = \frac{5}{20}$
$\frac{3}{5} = \frac{}{20}$ $\frac{3}{5} = \frac{3×4}{5×4} = \frac{12}{20}$
now, $\frac{5}{20}$ and $\frac{12}{20}$ have like denominators.
find the lcm of the denominators. then, rename the fractions so they have like denominators.

1. $\frac{1}{4}$ and $\frac{2}{3}$

4: _____________
3: _____________
$\frac{□}{□}$ and $\frac{□}{□}$

1. $\frac{3}{8}$ and $\frac{7}{10}$

8: _____________
10: _____________
$\frac{□}{□}$ and $\frac{□}{□}$

1. $\frac{4}{7}$ and $\frac{2}{3}$

7: _____________
3: _____________
$\frac{□}{□}$ and $\frac{□}{□}$

1. $\frac{2}{5}$ and $\frac{3}{4}$

5: _____________
4: _____________
$\frac{□}{□}$ and $\frac{□}{□}$

1. $\frac{1}{9}$ and $\frac{2}{3}$

9: _____________
3: _____________
$\frac{□}{□}$ and $\frac{□}{□}$

1. $\frac{5}{8}$ and $\frac{3}{16}$

8: _____________
16: _____________
$\frac{□}{□}$ and $\frac{□}{□}$

Explanation:

Problem 1: $\boldsymbol{\frac{1}{4}}$ and $\boldsymbol{\frac{2}{3}}$

Step1: Find multiples of 4 and 3

Multiples of 4: $4, 8, 12, 16, \dots$
Multiples of 3: $3, 6, 9, 12, \dots$
LCM of 4 and 3 is 12.

Step2: Rename $\frac{1}{4}$ with denominator 12

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

Step3: Rename $\frac{2}{3}$ with denominator 12

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

Problem 2: $\boldsymbol{\frac{3}{8}}$ and $\boldsymbol{\frac{7}{10}}$

Step1: Find multiples of 8 and 10

Multiples of 8: $8, 16, 24, 32, 40, \dots$
Multiples of 10: $10, 20, 30, 40, \dots$
LCM of 8 and 10 is 40.

Step2: Rename $\frac{3}{8}$ with denominator 40

$\frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40}$

Step3: Rename $\frac{7}{10}$ with denominator 40

$\frac{7}{10} = \frac{7 \times 4}{10 \times 4} = \frac{28}{40}$

Problem 3: $\boldsymbol{\frac{4}{7}}$ and $\boldsymbol{\frac{2}{3}}$

Step1: Find multiples of 7 and 3

Multiples of 7: $7, 14, 21, 28, \dots$
Multiples of 3: $3, 6, 9, 12, 15, 18, 21, \dots$
LCM of 7 and 3 is 21.

Step2: Rename $\frac{4}{7}$ with denominator 21

$\frac{4}{7} = \frac{4 \times 3}{7 \times 3} = \frac{12}{21}$

Step3: Rename $\frac{2}{3}$ with denominator 21

$\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}$

Problem 4: $\boldsymbol{\frac{2}{5}}$ and $\boldsymbol{\frac{3}{4}}$

Answer:

s:

1. Multiples of 4: $4, 8, 12, 16, \dots$; Multiples of 3: $3, 6, 9, 12, \dots$; $\frac{3}{12}$ and $\frac{8}{12}$
2. Multiples of 8: $8, 16, 24, 32, 40, \dots$; Multiples of 10: $10, 20, 30, 40, \dots$; $\frac{15}{40}$ and $\frac{28}{40}$
3. Multiples of 7: $7, 14, 21, 28, \dots$; Multiples of 3: $3, 6, 9, 12, 15, 18, 21, \dots$; $\frac{12}{21}$ and $\frac{14}{21}$
4. Multiples of 5: $5, 10, 15, 20, \dots$; Multiples of 4: $4, 8, 12, 16, 20, \dots$; $\frac{8}{20}$ and $\frac{15}{20}$
5. Multiples of 9: $9, 18, 27, \dots$; Multiples of 3: $3, 6, 9, 12, \dots$; $\frac{1}{9}$ and $\frac{6}{9}$
6. Multiples of 8: $8, 16, 24, \dots$; Multiples of 16: $16, 32, 48, \dots$; $\frac{10}{16}$ and $\frac{3}{16}$