# 1.6 Add and subtract fractions  (Page 2/4)

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When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.

Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need. In [link] , the LCD, 36, has two factors of 2 and two factors of $3.$

The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.

The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.

We will apply this method as we subtract the fractions in [link] .

Subtract: $\frac{7}{15}-\phantom{\rule{0.2em}{0ex}}\frac{19}{24}.$

## Solution

Do the fractions have a common denominator? No, so we need to find the LCD.

 Find the LCD. $\phantom{\rule{5em}{0ex}}$ Notice, 15 is “missing” three factors of 2 and 24 is “missing” the 5 from the factors of the LCD. So we multiply 8 in the first fraction and 5 in the second fraction to get the LCD. Rewrite as equivalent fractions with the LCD. Simplify. Subtract. $\phantom{\rule{2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{39}{120}$ Check to see if the answer can be simplified. $\phantom{\rule{2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{13\cdot 3}{40\cdot 3}$ Both 39 and 120 have a factor of 3. Simplify. $\phantom{\rule{2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{13}{40}$

Do not simplify the equivalent fractions! If you do, you’ll get back to the original fractions and lose the common denominator!

Subtract: $\frac{13}{24}-\phantom{\rule{0.2em}{0ex}}\frac{17}{32}.$

$\frac{1}{96}$

Subtract: $\frac{21}{32}-\phantom{\rule{0.2em}{0ex}}\frac{9}{28}.$

$\frac{75}{224}$

In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.

Add: $\frac{3}{5}+\frac{x}{8}.$

## Solution

The fractions have different denominators.

 $\phantom{\rule{2em}{0ex}}$ Find the LCD. $\phantom{\rule{4em}{0ex}}$ Rewrite as equivalent fractions with the LCD. $\phantom{\rule{2em}{0ex}}$ Simplify. $\phantom{\rule{2em}{0ex}}$ Add. $\phantom{\rule{2em}{0ex}}$ Remember, we can only add like terms: 24 and 5 x are not like terms.

Add: $\frac{y}{6}+\frac{7}{9}.$

$\frac{9y+42}{54}$

Add: $\frac{x}{6}+\frac{7}{15}.$

$\frac{15x+42}{135}$

We now have all four operations for fractions. [link] summarizes fraction operations .

 Fraction Multiplication Fraction Division $\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}$ Multiply the numerators and multiply the denominators $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$ Multiply the first fraction by the reciprocal of the second. Fraction Addition Fraction Subtraction $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$ Add the numerators and place the sum over the common denominator. $\frac{a}{c}-\phantom{\rule{0.2em}{0ex}}\frac{b}{c}=\frac{a-b}{c}$ Subtract the numerators and place the difference over the common denominator. To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.

Simplify: $\frac{5x}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}$ $\frac{5x}{6}·\frac{3}{10}.$

## Solution

First ask, “What is the operation?” Once we identify the operation that will determine whether we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.

1. What is the operation? The operation is subtraction.
$\begin{array}{cccccc}\text{Do the fractions have a common denominator? No.}\hfill & & & & & \hfill \frac{5x}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}\hfill \\ \\ \\ \text{Rewrite each fraction as an equivalent fraction with the LCD.}\hfill & & & & & \hfill \begin{array}{c}\frac{5x·5}{6·5}-\phantom{\rule{0.2em}{0ex}}\frac{3·3}{10·3}\hfill \\ \phantom{\rule{0.6em}{0ex}}\frac{25x}{30}-\phantom{\rule{0.2em}{0ex}}\frac{9}{30}\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Subtract the numerators and place the difference over the}\hfill \\ \text{common denominators.}\hfill \end{array}\hfill & & & & & \hfill \frac{25x-9}{30}\hfill \\ \\ \\ \begin{array}{c}\text{Simplify, if possible There are no common factors.}\hfill \\ \text{The fraction is simplified.}\hfill \end{array}\hfill & & & & & \end{array}$
2. What is the operation? Multiplication.
$\begin{array}{cccccc}& & & & & \hfill \phantom{\rule{1em}{0ex}}\frac{5x}{6}·\frac{3}{10}\hfill \\ \\ \\ \begin{array}{c}\text{To multiply fractions, multiply the numerators and multiply}\hfill \\ \text{the denominators.}\hfill \end{array}\hfill & & & & & \hfill \phantom{\rule{1em}{0ex}}\frac{5x·3}{6·10}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite, showing common factors.}\hfill \\ \text{Remove common factors.}\hfill \end{array}\hfill & & & & & \hfill \phantom{\rule{1em}{0ex}}\frac{\overline{)5}x·\overline{)3}}{2·\overline{)3}·2·\overline{)5}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & & & \hfill \phantom{\rule{1em}{0ex}}\frac{x}{4}\hfill \end{array}$

Notice we needed an LCD to add $\frac{5x}{6}-\phantom{\rule{0.2em}{0ex}}\frac{3}{10},$ but not to multiply $\frac{5x}{6}·\frac{3}{10}.$

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
rectangular field solutions
What is this?
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the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
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Choli
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44.19%
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40.22%
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if you want the discounted price subtract $725 from$1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2% Orlando you could also just divide$725/$1299 and then subtract it from 1. then you get the same answer. Orlando p mulripied-5 and add 30 to it Tausif Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif How do you find divisible numbers without a calculator? Jacob Reply TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13 BAINAMA When she graduates college, Linda will owe$43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan? Ariana Reply Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus? Kirisma Reply 66miles/hour snigdha How did you work it out? Esther s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr Orlando hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! Heather look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number... Felix for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer... Felix —12 Niazmohammad Thanks Felix.l also get confused with signs. Esther Thank you for this Shatey ty Graham think about it like you lost$19 (-19), then found $7(+7). Totally you lost just$12 (-12)
Annushka
I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-$10 ... I also for a long time would draw it out on a number line to visualize it
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3
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