Calculate the LCM of two numbers and add/subtract fractions.

Question

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Here are the solutions to the problems:

A. Calculate the following

a)
Multiples of 7: 7, 14, 21, 28, 35, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, ...
LCM: $\boxed{\text{28}}$

b)
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, ...
Multiples of 8: 8, 16, 24, 32, 40, 48, ...
LCM: $\boxed{\text{40}}$

B. Calculate the following

a) $\frac{2}{3} + \frac{1}{2}$

Find the least common multiple (LCM) of the denominators 3 and 2.
The LCM of 3 and 2 is 6.
Convert the fractions to have a common denominator of 6.
$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$ $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$
Add the fractions.
$\frac{4}{6} + \frac{3}{6} = \frac{4+3}{6} = \frac{7}{6}$
The answer is $\boxed{\frac{7}{6}}$

b) $\frac{7}{8} - \frac{1}{2}$
Find the LCM of the denominators 8 and 2.
The LCM of 8 and 2 is 8.
Convert the fractions to have a common denominator of 8.
$\frac{7}{8}$ $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$
Subtract the fractions.
$\frac{7}{8} - \frac{4}{8} = \frac{7-4}{8} = \frac{3}{8}$
The answer is $\boxed{\frac{3}{8}}$

C. Calculate the following

a) $\frac{1}{2} + \frac{1}{3}$
Find the LCM of the denominators 2 and 3.
The LCM of 2 and 3 is 6.
Convert the fractions to have a common denominator of 6.
$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$ $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
Add the fractions.
$\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}$
The answer is $\boxed{\frac{5}{6}}$

b) $\frac{4}{5} - \frac{1}{2}$
Find the LCM of the denominators 5 and 2.
The LCM of 5 and 2 is 10.
Convert the fractions to have a common denominator of 10.
$\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$ $\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$
Subtract the fractions.
$\frac{8}{10} - \frac{5}{10} = \frac{8-5}{10} = \frac{3}{10}$
The answer is $\boxed{\frac{3}{10}}$

Calculate the LCM of two numbers and add/subtract fractions.

Step-by-step solution

Find the least common multiple (LCM) of the denominators 3 and 2.

Convert the fractions to have a common denominator of 6.

Add the fractions.

Find the LCM of the denominators 8 and 2.

Convert the fractions to have a common denominator of 8.

Subtract the fractions.

Find the LCM of the denominators 2 and 3.

Convert the fractions to have a common denominator of 6.

Add the fractions.

Find the LCM of the denominators 5 and 2.

Convert the fractions to have a common denominator of 10.

Subtract the fractions.

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