Equivalent Fractions

Study Sheet

Equivalent Fractions

Simplifying to lowest terms

Lowest Terms

Concept
The Big Idea
122448
Cut every piece in two: twice as many pieces, twice as many shaded, the same amount.

A fraction is in lowest terms when the top and bottom share no factor except 11. Divide both by their greatest common factor to get there in one step.

Two fractions are equivalent when they name the same amount of a whole, like 12\tfrac12 and 24\tfrac24: cutting each half into two pieces doubles the number of pieces and doubles the number you have, so nothing changes. Multiplying or dividing the top and bottom by the same number is the only move that keeps a fraction equal to itself.

Example
Simplifying 68\dfrac{6}{8}
6834
6 eighths and 3 quarters cover the same length.

Both share the factor 22: 6÷28÷2=34\dfrac{6 \div 2}{8 \div 2} = \dfrac{3}{4}. The picture shows the same shaded amount.

Example
Finding 23=?12\dfrac23 = \dfrac{?}{12}
23812
Each third split into 4 twelfths.

The bottom went from 33 to 1212, which is ×4\times 4. Cutting every third into 44 equal pieces makes twelfths, and each shaded third becomes 44 shaded twelfths, so the top is also ×4\times 4: 2×43×4=812\dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}.

Side note
Where it goes wrong

Adding the same number to the top and bottom. 12\tfrac{1}{2} is not 23\tfrac{2}{3} — adding 11 to each is not the same as cutting the pieces smaller. Only multiplying or dividing both parts by the same number keeps the value.

Try it
Try it: simplify 1216\tfrac{12}{16}

Write 1216\dfrac{12}{16} in lowest terms. What is the greatest factor that 1212 and 1616 share?

Show the answer

Factors of 1212: 1,2,3,4,6,121, 2, 3, 4, 6, 12. Factors of 1616: 1,2,4,8,161, 2, 4, 8, 16. The greatest shared factor is 44: 12÷416÷4=34\dfrac{12 \div 4}{16 \div 4} = \mathbf{\dfrac34}. Check: 33 and 44 share only the factor 11, so this is lowest terms.

In Plain Terms & More Examples

Concept
In plain terms

A fraction is in lowest terms when the top and bottom share no common factor but 11. Divide both by the biggest factor they share.

Example
Simplifying 812\dfrac{8}{12}
8 of 122 of 3
8 of 12 slices and 2 of 3 slices: the same part of the circle.

Both share 44: 8÷412÷4=23\dfrac{8 \div 4}{12 \div 4} = \dfrac{2}{3}.

Example
Simplifying 915\dfrac{9}{15}

Both share 33: 9÷315÷3=35\dfrac{9 \div 3}{15 \div 3} = \dfrac{3}{5}.

Side note
Where it goes wrong

Stopping too soon. Dividing 812\dfrac{8}{12} by 22 gives 46\dfrac{4}{6}. That is equal, but not in lowest terms, because 44 and 66 still share the factor 22. Divide again to get 23\dfrac{2}{3}, or divide by the greatest common factor, 44, in one step.

Try it
Try it: simplify 1421\tfrac{14}{21}

Write 1421\dfrac{14}{21} in lowest terms.

Show the answer

14=2×714 = 2 \times 7 and 21=3×721 = 3 \times 7, so they share 77: 14÷721÷7=23\dfrac{14 \div 7}{21 \div 7} = \mathbf{\dfrac23}.

Comparing Fractions

Concept
Same bottom, then compare tops
2381234912
In twelfths the two fractions are 8/12 and 9/12: one twelfth apart.

To compare 23\tfrac23 and 34\tfrac34, rewrite both with denominator 1212: 812\tfrac{8}{12} and 912\tfrac{9}{12}. Now the tops decide: 34\tfrac34 is bigger.

Example
Ordering 12,35,410\tfrac12, \tfrac35, \tfrac{4}{10}
011021031041051061071081091014101235
On one line, left means smaller: 4/10, then 1/2 = 5/10, then 3/5 = 6/10.

Use tenths: 510,610,410\tfrac{5}{10}, \tfrac{6}{10}, \tfrac{4}{10}. From least to greatest: 410,12,35\tfrac{4}{10}, \tfrac12, \tfrac35.

Example
Same top, different bottoms: 35\tfrac35 or 38\tfrac38?
3538
Three big pieces beat three small pieces.

Both fractions count 33 pieces. A whole cut into 55 parts has bigger parts than one cut into 88, so 33 fifths is more than 33 eighths: 35>38\dfrac35 > \dfrac38. When the tops match, no common denominator is needed.

Side note
Where it goes wrong

Thinking a bigger denominator means a bigger fraction. 18\dfrac18 is smaller than 14\dfrac14: the more pieces you cut a whole into, the smaller each piece is. In eighths, 14=28\dfrac14 = \dfrac28, which is more than 18\dfrac18.

Try it
Try it: which is larger?

Which is larger, 58\tfrac{5}{8} or 712\tfrac{7}{12}?

Show the answer

Common denominator 2424: 58=1524\tfrac{5}{8} = \tfrac{15}{24} and 712=1424\tfrac{7}{12} = \tfrac{14}{24}. So 58\mathbf{\tfrac58} is larger.

Word Problems

Concept
Same-size wholes

Fractions of different things cannot be compared. Before comparing, check that the wholes are the same size: the same size pizza, the same length of ribbon, the same class.

12\tfrac12 of a large pizza is more food than 12\tfrac12 of a small one, even though the fractions are equal. A fraction tells you what part of its own whole you have. That is why word problems say "a same-size pizza": only then do the fractions alone decide.

Example
Two pizzas
AnaBen
Ana: 3 of 8 slices. Ben: 2 of 6 slices of a same-size pizza.

Ana ate 38\tfrac{3}{8} of a pizza and Ben ate 26\tfrac{2}{6} of a same-size pizza. 38=924\tfrac{3}{8} = \tfrac{9}{24} and 26=824\tfrac{2}{6} = \tfrac{8}{24}, so Ana ate slightly more.

Example
Equal recipes

Jo uses 24\tfrac24 cup of milk and Kim uses 48\tfrac48 cup. Who uses more? Multiply the top and bottom of 24\tfrac24 by 22: 2×24×2=48\tfrac{2 \times 2}{4 \times 2} = \tfrac48. They use exactly the same amount.

Side note
Where it goes wrong

Comparing parts of different wholes. Mia ate 14\tfrac14 of a large cake and Tom ate 12\tfrac12 of a cupcake. 12\tfrac12 is more than 14\tfrac14, but Mia ate far more cake. The fractions decide only when the wholes match.

Try it
Try it: the garden

In a garden 69\tfrac{6}{9} of the beds have tomatoes. Write that fraction in lowest terms.

Show the answer

The GCF of 66 and 99 is 33: 6÷39÷3=23\tfrac{6\div3}{9\div3} = \mathbf{\tfrac23}.

Fractions on a Number Line

Concept
One point, many names

On a number line from 00 to 11, a fraction is a point. Equivalent fractions are different names for the same point: 34\tfrac34 and 68\tfrac68 land in exactly the same place.

To mark 34\tfrac34, cut the space from 00 to 11 into 44 equal jumps and count 33 of them. To mark 68\tfrac68, cut the same space into 88 equal jumps (each quarter split in two) and count 66. Six small jumps cover the same distance as three big ones, so the two fractions sit at one point.

Example
Marking 34\tfrac34 and 68\tfrac68
014243410182838485868781
3 big jumps and 6 small jumps end at the same point.

Quarters: cut 00 to 11 into 44 equal jumps and count 33. Eighths: cut it into 88 jumps and count 66. Both counts stop at the same point, so 34=68\dfrac34 = \dfrac68.

Example
Where is 12\tfrac12 on a line marked in sixths?
01626364656112
Halfway is the third mark of six.

Halfway from 00 to 11 is 33 of the 66 jumps: 12=36\dfrac12 = \dfrac36. Check with equivalent fractions: 1×32×3=36\dfrac{1 \times 3}{2 \times 3} = \dfrac36.

Example
Whole numbers as fractions
0313233343536312
Three thirds make 1; six thirds make 2.

On a line marked in thirds, count the jumps to 22. There are 33 jumps in each whole, so 66 jumps in two wholes: 2=632 = \dfrac63. In the same way 1=331 = \dfrac33. Any fraction whose top equals its bottom is one whole.

Side note
Where it goes wrong

Counting tick marks instead of jumps. A line from 00 to 11 with 55 tick marks (including the ones at 00 and 11) has only 44 jumps, so it is marked in fourths, not fifths. Count the spaces between the ticks.

Try it
Try it: the tenths line
01
Ten equal jumps from 0 to 1.

A number line from 00 to 11 is marked in tenths. Which mark is the same point as 25\tfrac25?

Show the answer

Each fifth is 22 tenths (15=210\tfrac15 = \tfrac{2}{10}), so 25=2×25×2=410\tfrac25 = \tfrac{2 \times 2}{5 \times 2} = \mathbf{\tfrac{4}{10}}: the fourth mark after 00.

Why It Works

Multiplying the numerator and denominator by the same number kk is multiplying by kk=1\tfrac kk = 1, and multiplying by 11 changes nothing: ab⋅kk=akbk\tfrac{a}{b}\cdot\tfrac{k}{k} = \tfrac{ak}{bk}. In a picture, akbk\tfrac{ak}{bk} is the same shaded region as ab\tfrac ab with each piece cut into kk thinner slices. Reducing to lowest terms runs the same idea backwards, dividing out every common factor until the greatest common factor of the top and bottom is 11.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.