4.5 Add and subtract fractions with different denominators  (Page 5/7)

Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

Evaluate variable expressions with fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Evaluate $x+\frac{1}{3}$ when

1. ⓐ $x=-\frac{1}{3}$
2. ⓑ $x=-\frac{3}{4}.$

Solution

ⓐ To evaluate $x+\frac{1}{3}$ when $x=-\frac{1}{3},$ substitute $-\frac{1}{3}$ for $x$ in the expression.

	$x+\frac{1}{3}$
Simplify.	$0$

ⓑ To evaluate $x+\frac{1}{3}$ when $x=-\frac{3}{4},$ we substitute $-\frac{3}{4}$ for $x$ in the expression.

	$x+\frac{1}{3}$
Rewrite as equivalent fractions with the LCD, 12.	$-\frac{3·3}{4·3}+\frac{1·4}{3·4}$
Simplify the numerators and denominators.	$-\frac{9}{12}+\frac{4}{12}$
Add.	$-\frac{5}{12}$

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Evaluate $y-\frac{5}{6}$ when $y=-\frac{2}{3}.$

Solution

We substitute $-\frac{2}{3}$ for $y$ in the expression.

	$y-\frac{5}{6}$
Rewrite as equivalent fractions with the LCD, 6.	$-\frac{4}{6}-\frac{5}{6}$
Subtract.	$-\frac{9}{6}$
Simplify.	$-\frac{3}{2}$

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Evaluate $2x^{2}y$ when $x=\frac{1}{4}$ and $y=-\frac{2}{3}.$

Solution

Substitute the values into the expression. In $2x^{2}y,$ the exponent applies only to $x.$

Simplify exponents first.
Multiply. The product will be negative.
Simplify.
Remove the common factors.
Simplify.

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Evaluate $\frac{p+q}{r}$ when $p=\mathrm{-4},q=\mathrm{-2},$ and $r=8.$

Solution

We substitute the values into the expression and simplify.

	$\frac{p+q}{r}$
Add in the numerator first.	$-\frac{6}{8}$
Simplify.	$-\frac{3}{4}$

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Key concepts

• Find the least common denominator (LCD) of two fractions.
  1. Factor each denominator into its primes.
  2. List the primes, matching primes in columns when possible.
  3. Bring down the columns.
  4. Multiply the factors. The product is the LCM of the denominators.
  5. The LCM of the denominators is the LCD of the fractions.
• Equivalent Fractions Property
  • If $a,b$ , and $c$ are whole numbers where $b\ne 0$ , $c\ne 0$ then
    $\frac{a}{b}=\frac{a\cdot c}{b\cdot c}$ and $\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
• Convert two fractions to equivalent fractions with their LCD as the common denominator.
  1. Find the LCD.
  2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
  3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
  4. Simplify the numerator and denominator.
• Add or subtract fractions with different denominators.
  1. Find the LCD.
  2. Convert each fraction to an equivalent form with the LCD as the denominator.
  3. Add or subtract the fractions.
  4. Write the result in simplified form.
• Summary of Fraction Operations
  • Fraction multiplication: Multiply the numerators and multiply the denominators.
    $\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$
  • Fraction division: Multiply the first fraction by the reciprocal of the second.
    $\frac{a}{b}+\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$
  • Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
    $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$
  • Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
    $\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}$
• Simplify complex fractions.
  1. Simplify the numerator.
  2. Simplify the denominator.
  3. Divide the numerator by the denominator.
  4. Simplify if possible.