Question

lets learn!
to add fractions with different denominators, you can make equivalent fractions using the least common denominator (lcd). the lcd is the smallest common multiple of both denominators.
try it for \\(\frac{1}{4}+\frac{3}{5}\\).
to find the lcd, write a few multiples of each denominator. circle the smallest multiple they share. here, the lcd is 20.
multiples of 4: 4, 8, 12, 16,
multiples of 5: 5, 10, 15, 20
make equivalent fractions using the lcd as the denominator, and then add.
\\(\

$$\begin{align*}\\frac{1}{4} + \\frac{3}{5} &= \\\\ \\frac{5}{20} + \\frac{12}{20} &= \\frac{17}{20}\\end{align*}$$

\\)
add. write your answer in simplest form.
\\(\frac{2}{3}+\frac{1}{4}=\frac{11}{12}\\)
\\(\frac{8}{12}+\frac{3}{12}=\frac{11}{12}\\)
\\(\frac{1}{5}+\frac{3}{10}=\underline{\quad\quad}\\)
\\(\frac{1}{2}+\frac{2}{7}=\underline{\quad\quad}\\)
\\(\frac{1}{6}+\frac{2}{9}=\underline{\quad\quad}\\)

Explanation:

1. Solve $\boldsymbol{\frac{1}{5} + \frac{3}{10}}$

Step1: Find the LCD of 5 and 10

The multiples of 5 are $5, 10, 15, \dots$ and the multiples of 10 are $10, 20, 30, \dots$. The least common denominator (LCD) is 10.

Step2: Make equivalent fractions

For $\frac{1}{5}$, multiply numerator and denominator by 2: $\frac{1\times2}{5\times2}=\frac{2}{10}$. $\frac{3}{10}$ remains as it is.

Step3: Add the fractions

$\frac{2}{10}+\frac{3}{10}=\frac{2 + 3}{10}=\frac{5}{10}$. Simplify by dividing numerator and denominator by 5: $\frac{5\div5}{10\div5}=\frac{1}{2}$.

Step1: Find the LCD of 2 and 7

The multiples of 2 are $2, 4, 6, 8, 10, 12, 14, \dots$ and the multiples of 7 are $7, 14, 21, \dots$. The LCD is 14.

Step2: Make equivalent fractions

For $\frac{1}{2}$, multiply numerator and denominator by 7: $\frac{1\times7}{2\times7}=\frac{7}{14}$. For $\frac{2}{7}$, multiply numerator and denominator by 2: $\frac{2\times2}{7\times2}=\frac{4}{14}$.

Step3: Add the fractions

$\frac{7}{14}+\frac{4}{14}=\frac{7 + 4}{14}=\frac{11}{14}$.

Step1: Find the LCD of 6 and 9

The multiples of 6 are $6, 12, 18, 24, \dots$ and the multiples of 9 are $9, 18, 27, \dots$. The LCD is 18.

Step2: Make equivalent fractions

For $\frac{1}{6}$, multiply numerator and denominator by 3: $\frac{1\times3}{6\times3}=\frac{3}{18}$. For $\frac{2}{9}$, multiply numerator and denominator by 2: $\frac{2\times2}{9\times2}=\frac{4}{18}$.

Step3: Add the fractions

$\frac{3}{18}+\frac{4}{18}=\frac{3 + 4}{18}=\frac{7}{18}$.

Answer:

$\frac{1}{2}$

2. Solve $\boldsymbol{\frac{1}{2} + \frac{2}{7}}$