Before you read

Answer first — see what you already know.

Emma saves half of each paycheck plus $$3$ from tips. If the total saved is $$10$ , the equation $\frac{x}{2} + 3 = 10$ gives the paycheck amount $x$ . What is $x$ ?

A
$7$
B
$14$
C
$20$
D
$\frac{13}{2}$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A recipe requires $\frac{x}{3}$ cups of flour from one bag and $\frac{x}{6}$ cups from another. If the total flour used is $5$ cups, what is $x$ ? ( $\frac{x}{3} + \frac{x}{6} = 5$ )

A
$10$
B
$15$
C
$6$
D
$30$

Check your answer

Answer:

Theory

Clearing Fractions: Multiply by the LCD

Equations with fractions look scary, but there's a simple trick: multiply every term by the Least Common Denominator (LCD). This clears all fractions at once.

The LCD is the smallest number that all denominators divide into evenly.

Examples of LCDs:
- Denominators $2$ and $3$ $\to$ LCD = $6$
- Denominators $4$ and $6$ $\to$ LCD = $12$
- Denominators $3$ , $4$ , and $6$ $\to$ LCD = $12$

$x/2 + 3 = 10$
$x + 6 = 20$ multiply by LCD (2)
$x = 14$ subtract 6

Clearing fractions by multiplying by the LCD

Example 1

Solve $\frac{x}{3} + \frac{x}{4} = 7$

Step 1: Find the LCD of $3$ and $4$ : LCD = $12$

Step 2: Multiply every term by $12$ :

$12 \cdot \frac{x}{3} + 12 \cdot \frac{x}{4} = 12 \cdot 7$

$4x + 3x = 84$

Step 3: Combine and solve:

$7x = 84$

$x = 12$

x = 12

\frac{12}{3} + \frac{12}{4} = 4 + 3 = 7

✓

Example 2

Solve $\frac{2x + 1}{5} = 3$

Step 1: Multiply both sides by $5$ :

$2x + 1 = 15$

Step 2: Subtract $1$ :

$2x = 14$

Step 3: Divide by $2$ :

$x = 7$

x = 7

\frac{2(7) + 1}{5} = \frac{15}{5} = 3

✓

Theory

Clearing Decimals: Multiply by a Power of 10

To clear decimals, multiply every term by $10$ , $100$ , or $1000$ depending on the most decimal places in the equation.

- 1 decimal place $\to$ multiply by $10$
- 2 decimal places $\to$ multiply by $100$
- 3 decimal places $\to$ multiply by $1000$

Example 1

Solve $0.3x + 1.5 = 4.2$

Step 1: Multiply every term by $10$ (one decimal place):

$3x + 15 = 42$

Step 2: Subtract $15$ :

$3x = 27$

Step 3: Divide by $3$ :

$x = 9$

x = 9

0.3(9) + 1.5 = 2.7 + 1.5 = 4.2

✓

Example 2

Solve $0.25x - 0.10 = 0.40$

Step 1: Multiply every term by $100$ (two decimal places):

$25x - 10 = 40$

Step 2: Add $10$ :

$25x = 50$

Step 3: Divide by $25$ :

$x = 2$

x = 2

0.25(2) - 0.10 = 0.50 - 0.10 = 0.40

✓

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