Many names, one number

The idea

A rational number has many equivalent forms p/q: dividing numerator and denominator by a common factor gives an equivalent one, and p/q is normally taken with p and q co-prime.

Many names

The number −1/3 can be written as −2/6, or −3/9, or −10/30, and on and on. A rational number has many names.

Do 12/30 and 2/5 name the same number?

• Yes
• No
• Not sure

Cut and join

The shaded part of the bar never changes. Divide the top and the bottom by a common factor, or multiply both by the same number.

Lowest terms

A rational number has many equivalent forms p/q. Dividing the numerator and the denominator by a common factor gives an equivalent form. We normally write p/q with p and q co-prime: no common factor except 1.

Dividing 12 and 30 by 6 gives 2/5. The bar shows the same shaded part. 2 and 5 have no common factor except 1, so 2/5 is in lowest terms.

Notes

A rational number has many equivalent forms. Divide the top and the bottom by a common factor to get an equivalent one. Lowest terms: p and q are co-prime.

Check yourself

Write 18/24 in lowest terms. What is the numerator?

Answer: 3

18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/24 = 3/4. 3 and 4 share no factor except 1.

Which fraction is already in lowest terms?

Which fraction is not equal to 1/2?

“p and q are co-prime.” What does that mean?

• 4/9 — correct. Yes. 4 and 9 have no common factor except 1.
• 6/9. 6 and 9 are both divisible by 3. Divide by 3 to get 2/3.
• 10/15. 10 and 15 are both divisible by 5. Divide by 5 to get 2/3.
• 3/6. 3/6 = 1/2, because 3 × 2 = 6.
• 5/10. 5/10 = 1/2, because 5 × 2 = 10.
• 4/6 — correct. 4/6 = 2/3, which is not 1/2.
• They have no common factor except 1 — correct. Yes. 4 and 9 are co-prime even though neither is prime.
• They are both prime numbers. Co-prime is different: 4 and 9 are co-prime but neither is a prime number.
• They are equal. If p and q were equal, p/q would simply be 1.