A fraction is more than a slice of pizza. It is a number, it can be a division, and you can multiply it and divide it just like whole numbers. In this chapter you will learn to see each operation in a picture first, then turn the picture into a short, reliable rule.
1. Fractions as division
The fraction \(\dfrac{a}{b}\) is the answer to \(a \div b\) (with \(b\) not zero). The numerator is the amount being shared and the denominator is the number of equal shares.
Imagine 3 granola bars shared equally among 4 hikers. Cut each bar into 4 equal pieces: that makes 12 pieces, and each hiker takes 3 of them. Each hiker gets \(\dfrac{3}{4}\) of a bar, so \(3 \div 4 = \dfrac{3}{4}\).
When the numerator is larger than the denominator, the quotient is larger than 1. For example, \(7 \div 2 = \dfrac{7}{2} = 3\tfrac{1}{2}\): if 7 pancakes are shared by 2 people, each person gets 3 whole pancakes and half of another one.
Five friends share 2 quarts of lemonade equally. Each friend gets \(2 \div 5 = \dfrac{2}{5}\) of a quart.
2. Multiplying a whole number by a fraction
Multiplying a whole number by a fraction is repeated addition: \(4 \times \dfrac{2}{5}\) means four groups of \(\dfrac{2}{5}\). Add them: \(\dfrac{2}{5}+\dfrac{2}{5}+\dfrac{2}{5}+\dfrac{2}{5} = \dfrac{8}{5}\).
- Multiply the whole number by the numerator.
- Keep the denominator the same.
- Simplify, or write the answer as a mixed number.
A hiking trail loop is \(\dfrac{3}{8}\) mile long. You walk it 6 times. Distance: \(6 \times \dfrac{3}{8} = \dfrac{18}{8} = \dfrac{9}{4} = 2\tfrac{1}{4}\) miles.
3. Multiplying two fractions
\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d} \]
Multiply the numerators together and the denominators together.
The word “of” often means multiply. Finding \(\dfrac{2}{5}\) of \(\dfrac{3}{4}\) is the same as computing \(\dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{6}{20} = \dfrac{3}{10}\).
You can save work by simplifying before you multiply. In \(\dfrac{3}{4} \times \dfrac{8}{9}\), the 4 and the 8 share the factor 4, and the 3 and the 9 share the factor 3. So \(\dfrac{3}{4} \times \dfrac{8}{9} = \dfrac{1 \times 2}{1 \times 3} = \dfrac{2}{3}\).
A garden is \(\dfrac{3}{5}\) planted with vegetables, and \(\dfrac{1}{2}\) of the vegetable part is beans. Beans cover \(\dfrac{1}{2} \times \dfrac{3}{5} = \dfrac{3}{10}\) of the garden.
4. The area model
A picture explains why the rule works. Draw a unit square. To show \(\dfrac{2}{3} \times \dfrac{3}{4}\), cut it into 3 equal columns and 4 equal rows. That gives \(3 \times 4 = 12\) small rectangles of the same size. Shade 2 columns, then shade 3 rows. The overlap holds \(2 \times 3 = 6\) small rectangles out of 12.
So \(\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{6}{12} = \dfrac{1}{2}\). The denominators multiply because the cuts create \(3 \times 4\) pieces, and the numerators multiply because the overlap is \(2\) columns by \(3\) rows.
The same idea gives the area of a rectangle with fraction side lengths: a rectangle \(\dfrac{3}{4}\) foot by \(\dfrac{2}{3}\) foot has area \(\dfrac{1}{2}\) square foot.
5. Multiplication as scaling
Multiplying by a fraction stretches or shrinks a number. You can predict the size of a product without calculating it.
- If the factor is greater than 1, the product is greater than the other number.
- If the factor equals 1 (such as \(\dfrac{7}{7}\)), the product stays the same.
- If the factor is less than 1, the product is smaller than the other number.
On my home planet we check every answer with scaling first. If you multiply 40 by \(\dfrac{3}{5}\) and get more than 40, something went wrong before you even picked up a pencil!
6. Multiplying mixed numbers
- Change each mixed number into an improper fraction.
- Multiply the numerators and the denominators.
- Simplify and, if you like, change back to a mixed number.
Compute \(2\tfrac{1}{2} \times 1\tfrac{1}{5}\). Convert: \(2\tfrac{1}{2} = \dfrac{5}{2}\) and \(1\tfrac{1}{5} = \dfrac{6}{5}\). Then \(\dfrac{5}{2} \times \dfrac{6}{5} = \dfrac{30}{10} = 3\).
Do not multiply the whole parts and the fraction parts separately. \(2\tfrac{1}{2} \times 1\tfrac{1}{5}\) is not \(2 \times 1\) and \(\dfrac{1}{2} \times \dfrac{1}{5}\) put side by side. Always convert to improper fractions first.
7. Dividing with unit fractions
A unit fraction has 1 as its numerator, such as \(\dfrac{1}{3}\) or \(\dfrac{1}{8}\). Grade 5 works with two kinds of division involving unit fractions.
A unit fraction divided by a whole number
Splitting \(\dfrac{1}{2}\) into 3 equal parts makes pieces that are each \(\dfrac{1}{6}\) of the whole.
So \(\dfrac{1}{2} \div 3 = \dfrac{1}{2 \times 3} = \dfrac{1}{6}\). In general, \(\dfrac{1}{b} \div n = \dfrac{1}{b \times n}\).
A whole number divided by a unit fraction
Dividing asks “how many pieces fit?” With 4 wholes cut into thirds, there are \(4 \times 3 = 12\) pieces. So \(4 \div \dfrac{1}{3} = 12\).
\(\dfrac{1}{b} \div n = \dfrac{1}{b \times n}\) and \(n \div \dfrac{1}{b} = n \times b\).
A ribbon \(\dfrac{1}{4}\) yard long is cut into 3 equal pieces: each piece is \(\dfrac{1}{4} \div 3 = \dfrac{1}{12}\) yard. How many \(\dfrac{1}{6}\)-pound scoops fill a 5-pound bag? \(5 \div \dfrac{1}{6} = 30\) scoops.
You can always check a division with multiplication: since \(12 \times \dfrac{1}{3} = 4\), we know \(4 \div \dfrac{1}{3} = 12\).
8. Fraction word problems
- Read the problem and say in your own words what is asked.
- Decide the operation: sharing or “how many fit” means divide, “of” or “groups of” means multiply.
- Estimate with scaling, then calculate and simplify.
- Write a sentence with the unit.
A pool is \(3\tfrac{1}{2}\) feet deep at one end. A float line is at \(\dfrac{2}{7}\) of that depth. Its depth is \(\dfrac{2}{7} \times \dfrac{7}{2} = 1\) foot. The float line sits 1 foot below the surface.
Key takeaways
- \(\dfrac{a}{b} = a \div b\): a fraction is a division.
- To multiply fractions, multiply numerators and multiply denominators; simplify at the end or before.
- The area model shows \(\dfrac{a}{b} \times \dfrac{c}{d}\) as the overlap of shaded columns and rows.
- A factor greater than 1 makes a product bigger; a factor less than 1 makes it smaller.
- Convert mixed numbers to improper fractions before multiplying.
- \(\dfrac{1}{b} \div n = \dfrac{1}{b \times n}\) and \(n \div \dfrac{1}{b} = n \times b\).
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Speed drill for Grade 5: how many in 60 seconds?
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