Simplifying never changes how much a fraction is worth. It only changes the size of the pieces you are counting in.
Two fractions can look completely different and still name the same amount. Twelve eighteenths of a chocolate bar and two thirds of the same bar are the same quantity of chocolate, described with different sized pieces.
Turning one into the other is the skill. It rests on a single move, and on knowing exactly when that move is allowed.
1. The same amount, different names
Shade the same portion of three identical bars, then cut each bar into a different number of pieces. The shading never moves. Only the count of pieces and the size of each piece change.
Figure 1. The shaded edge lands in the same place on all three bars, which is what makes these three fractions equivalent rather than merely similar.
2. Divide the top and the bottom by the same number
Dividing both parts of a fraction by the same number regroups the pieces without changing the amount. Skill 0.08 used the same principle on an equation: do it to both parts or the value shifts.
Figure 2. Six is the largest number that divides both 12 and 18, so one step reaches simplest form. A smaller shared factor would work too, just slower.
Key idea
Divide the numerator and the denominator by the same number. A fraction is in simplest form when no number above 1 divides both.
3. The mistake: canceling terms instead of factors
Canceling is really division, so it can only remove something that divides the whole numerator and the whole denominator. A number joined to others by a plus sign is a term, and it divides only itself. Striking it out removes part of the top and all of the bottom, which is not one division.
Watch out: a term is not a factor
The same digit, struck out the same way, is legal on the right and illegal on the left.
Sense check: before canceling, ask whether the number multiplies everything around it. If a plus or minus sign separates it from its neighbor, it cannot go.
4. Writing both parts as products first
The safest route is to rewrite the numerator and denominator as products that show the shared factor, then remove it. Writing 84 as 12 × 7 and 144 as 12 × 12 makes the canceling obvious and legal.
Figure 3. If you cannot see the largest shared factor at once, remove a smaller one and repeat. Two easy steps beat one guessed step.
5. Recognizing simplest form
A fraction is in simplest form when the only number dividing both parts is 1. Checking the small primes in order settles most cases quickly.
Figure 4. The last row is the stopping condition. When the only shared factor is 1, there is nothing left to remove.
6. Worked examples
Example A
Simplify 18 over 24.
Both are even, and both divide by 6. Write 18 as 6 × 3 and 24 as 6 × 4, then remove the shared 6.
18 over 24 simplifies to 3 over 4. Sense check: 3 and 4 share no factor above 1, so this is simplest form.
Example B
Simplify 45 over 60.
If 15 is not obvious, take it in stages. Divide both by 5 to reach 9 over 12, then divide both by 3.
45 over 60 simplifies to 3 over 4. Sense check: two small steps landed on the same answer one step of 15 would have given.
Example C
A Riverside class of 30 students has 18 signed up for the science fair. Write that share in simplest form.
The fraction is 18 over 30. Both divide by 6, giving 3 over 5.
Three fifths of the class signed up. Sense check: 3 out of every 5 students is 18 out of 30, since 6 groups of 5 make 30.
Words to know
Numerator
The top number, counting how many pieces you have.
Denominator
The bottom number, naming how many pieces make a whole.
Equivalent
Naming the same amount with different numbers.
Factor
A number multiplied by another; only factors may be canceled.
Simplest form
When the only number dividing both parts is 1.
7. A look ahead
Coming next: Skill 0.10, adding, subtracting, multiplying and dividing fractions
Skill 0.10 puts fractions through all four operations, where each one behaves differently. Two traps wait there: adding the denominators as though they were counts, and flipping the wrong fraction when dividing.
Check your understanding
Explain why 12 over 18 and 2 over 3 name the same amount.
Explain why canceling the 3 in (3 + 4) over 3 is not allowed.
Describe how you know when a fraction is in simplest form.
Explain why dividing only the numerator would change the value.
Describe two different routes from 45 over 60 to simplest form.
What to carry away
Equivalent fractions name one amount using pieces of different sizes.
Divide the numerator and denominator by the same number to simplify.
Dividing only one part changes the value and breaks the equivalence.
Canceling is division, so only shared factors can be canceled.
A number joined by a plus or minus sign is a term, and terms never cancel.
Rewriting both parts as products makes a legal cancellation visible.
Several small steps reach the same simplest form as one large step.
A fraction is simplest when the only number dividing both parts is 1.