Pizza night, a recipe, a hike: fractions are everywhere! In this chapter you will break fractions into pieces, add them, subtract them, work with mixed numbers, and even multiply a fraction by a whole number. All the fractions you add or subtract here share the same bottom number, so the rules stay simple.
1. Decomposing fractions
To decompose a fraction means to break it into a sum of smaller fractions that have the same denominator. Every fraction is made of copies of a unit fraction, a fraction whose numerator is 1.
A unit fraction such as \(\dfrac{1}{8}\) is one part of a whole cut into equal parts. A fraction \(\dfrac{5}{8}\) is five copies of \(\dfrac{1}{8}\):
\[ \dfrac{5}{8}=\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8}+\dfrac{1}{8} = 5\times\dfrac{1}{8} \]
There are many ways to decompose the same fraction. The bar below shows \(\dfrac{5}{8}\) as \(\dfrac{2}{8}+\dfrac{3}{8}\).
Decompose \(\dfrac{7}{10}\) in two different ways.
\(\dfrac{7}{10}=\dfrac{3}{10}+\dfrac{4}{10}\) and also \(\dfrac{7}{10}=\dfrac{5}{10}+\dfrac{2}{10}\). The numerators add up to 7 and the denominator stays 10.
2. Adding fractions with like denominators
Fractions with like denominators are cut into the same size of parts, so you can simply count the parts.
Add the numerators and keep the denominator:
\[ \dfrac{a}{d}+\dfrac{b}{d}=\dfrac{a+b}{d} \]
Mia ran \(\dfrac{3}{10}\) of a mile, then walked \(\dfrac{4}{10}\) of a mile. How far did she go?
\(\dfrac{3}{10}+\dfrac{4}{10}=\dfrac{7}{10}\). Mia went \(\dfrac{7}{10}\) of a mile.
Never add the denominators! \(\dfrac{3}{10}+\dfrac{4}{10}\) is not \(\dfrac{7}{20}\). The parts are still tenths, so the denominator stays 10.
If the sum reaches the denominator, you have a whole: \(\dfrac{6}{6}=1\). For example \(\dfrac{2}{6}+\dfrac{4}{6}=\dfrac{6}{6}=1\).
3. Subtracting fractions with like denominators
Subtracting works the same way: you take parts away, and the size of the parts does not change.
Subtract the numerators and keep the denominator:
\[ \dfrac{a}{d}-\dfrac{b}{d}=\dfrac{a-b}{d} \quad (a\ge b) \]
A jug holds \(\dfrac{9}{12}\) of a gallon of juice. You pour out \(\dfrac{5}{12}\) of a gallon. How much is left?
\(\dfrac{9}{12}-\dfrac{5}{12}=\dfrac{4}{12}\) of a gallon is left.
To subtract a fraction from 1, write 1 as a fraction with the same denominator: \(1-\dfrac{3}{8}=\dfrac{8}{8}-\dfrac{3}{8}=\dfrac{5}{8}\).
4. Mixed numbers and improper fractions
A mixed number has a whole number part and a fraction part, like \(1\dfrac{3}{4}\). An improper fraction has a numerator greater than or equal to its denominator, like \(\dfrac{7}{4}\). They are two ways to write the same amount.
- Mixed number to improper fraction: multiply the whole number by the denominator, add the numerator, keep the denominator. For \(2\dfrac{3}{5}\): \(2\times5+3=13\), so \(\dfrac{13}{5}\).
- Improper fraction to mixed number: divide the numerator by the denominator. The quotient is the whole number and the remainder is the new numerator. For \(\dfrac{11}{4}\): \(11\div4=2\) remainder 3, so \(2\dfrac{3}{4}\).
The number line shows that \(\dfrac{7}{4}\) sits between 1 and 2, exactly three fourths of the way past 1.
Write \(\dfrac{17}{6}\) as a mixed number. \(17\div6=2\) remainder 5, so \(\dfrac{17}{6}=2\dfrac{5}{6}\).
5. Adding and subtracting mixed numbers
Handle the whole numbers and the fractions separately, then put them back together.
- Add the whole numbers.
- Add the fractions.
- If the fraction sum is an improper fraction, convert it and add its whole part to the total.
\(2\dfrac{5}{8}+1\dfrac{6}{8}\): wholes \(2+1=3\); fractions \(\dfrac{5}{8}+\dfrac{6}{8}=\dfrac{11}{8}=1\dfrac{3}{8}\). Total: \(3+1\dfrac{3}{8}=4\dfrac{3}{8}\).
To subtract, subtract the wholes and the fractions. If the first fraction is too small, regroup: borrow 1 whole and rewrite it as a fraction.
\(5\dfrac{1}{5}-2\dfrac{3}{5}\): \(\dfrac{1}{5}\) is smaller than \(\dfrac{3}{5}\), so write \(5\dfrac{1}{5}=4\dfrac{6}{5}\). Then \(4\dfrac{6}{5}-2\dfrac{3}{5}=2\dfrac{3}{5}\).
Check by converting: \(\dfrac{26}{5}-\dfrac{13}{5}=\dfrac{13}{5}=2\dfrac{3}{5}\).
On my planet we always check subtraction by adding back. If \(2\dfrac{3}{5}+2\dfrac{3}{5}\) does not give \(5\dfrac{1}{5}\) again, something is off. Try it!
6. Multiplying a fraction by a whole number
Multiplying a fraction by a whole number is repeated addition. Three groups of \(\dfrac{2}{5}\) is \(\dfrac{2}{5}+\dfrac{2}{5}+\dfrac{2}{5}\).
Multiply the whole number by the numerator and keep the denominator:
\[ n\times\dfrac{a}{d}=\dfrac{n\times a}{d} \]
\(4\times\dfrac{3}{8}=\dfrac{12}{8}=1\dfrac{4}{8}\).
7. Fraction word problems
- Read it and decide: are you joining parts, taking parts away, or repeating the same amount?
- Write the number sentence.
- Compute, convert to a mixed number if you like, and answer with a full sentence and a unit.
A baker uses \(\dfrac{3}{4}\) cup of sugar for one batch of cookies. How much sugar does she need for 5 batches?
\(5\times\dfrac{3}{4}=\dfrac{15}{4}=3\dfrac{3}{4}\). She needs \(3\dfrac{3}{4}\) cups of sugar.
Key takeaways
- A fraction is a sum of unit fractions: \(\dfrac{5}{8}=5\times\dfrac{1}{8}\).
- With like denominators, add or subtract the numerators and keep the denominator.
- Never add the denominators.
- Convert between improper fractions and mixed numbers with division and with multiplication.
- For mixed numbers, work on the wholes and the fractions, and regroup when needed.
- \(n\times\dfrac{a}{d}=\dfrac{n\times a}{d}\).
- In word problems, write a number sentence and answer with a unit.
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