Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Equation fixes the form, inequality bounds it

Find a fraction in lowest terms, with whole-number numerator and denominator, such that adding 3 to the denominator makes it equal 5/18 and adding 3 to the numerator makes it larger than 1/3.
ExpressionsNumber system
Your answer
How to solve
Strategy Convert to Algebra — The equation pins down the shape of the fraction up to one whole number k. The inequality then limits k to a few values, and the lowest-terms condition picks out the one that works.
1STEP 1

Name the fraction

Call the fraction y/x, where x and y are whole numbers with no common factor. The first condition says:

y/(x + 3) = 5/18, so 18y = 5(x + 3)
2STEP 2

Use that 5 and 18 share no factor

5 divides 18y but shares no factor with 18, so 5 divides y. Write y = 5k; then x + 3 = 18k for the same whole number k.

y = 5k, x = 18k - 3 (k = 1, 2, 3, …)
3STEP 3

Turn the second condition into an inequality in k

Adding 3 to the numerator must give more than 1/3. The denominator is positive, so clearing it keeps the direction.

(5k + 3)/(18k - 3) ＞ 1/3 → 15k + 9 ＞ 18k - 3 → 12 ＞ 3k → k ＜ 4
4STEP 4

List the candidates

Put k = 1, 2, 3 into 5k/(18k - 3).

5/15, 10/33, 15/51
5STEP 5

Keep the one in lowest terms

5/15 shares the factor 5 and 15/51 shares the factor 3, so both reduce. 10 and 33 share no factor other than 1.

gcd(5, 15) = 5, gcd(10, 33) = 1, gcd(15, 51) = 3
Answer
10/33
Adding 3 to the denominator: 10/36 = 5/18. Adding 3 to the numerator: 13/33 ＞ 11/33 = 1/3. Both conditions hold, and 10/33 cannot be reduced.
Takeaway

An equation tells you the shape of the answer, an inequality tells you how far to look, and lowest terms tells you which one to keep.

  • Name the fraction
  • Use that 5 and 18 share no factor
  • Turn the second condition into an inequality in k
  • List the candidates
  • Keep the one in lowest terms
Where next?
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▶ Practice — 12 problems