Numbers & Place Value

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

Turn leading digits into an inequality

A fraction x/y in lowest terms, less than 1, with 30 ＜ y ＜ 40, has 0 as the first digit after the decimal point and 6 as the second. Find the largest possible x + y.
Number systemExpressions
Your answer
How to solve
Strategy Convert to Algebra — Turn the digit condition into an inequality. Since y is trapped between 31 and 39, the inequality first forces the numerator, then gives a short list of denominators to test for lowest terms.
1STEP 1

Turn the digits into an inequality

A decimal that begins 0.06 is at least 0.06 and less than 0.07. Multiply through by y.

6/100 y ≤ x ＜ 7/100 y
2STEP 2

Pin down the numerator

With 31 ≤ y ≤ 39, the smallest the lower bound can be is 6/100 × 31 and the largest the upper bound can be is 7/100 × 39.

186/100 ≤ x ＜ 273/100
3STEP 3

Find the allowed denominators

Put x = 2 back: 0.06 ≤ 2/y ＜ 0.07 means 2/0.07 ＜ y ≤ 2/0.06.

200/7 ＜ y ≤ 100/3
4STEP 4

Keep lowest terms and maximize

2/32 reduces, so it is out. That leaves y = 31 and y = 33; the larger sum uses y = 33.

x + y = 2 + 33 = 35
Answer
35
Check: 2/33 = 0.060606… starts 0.06, and gcd(2, 33) = 1. The other survivor 2/31 = 0.0645… gives only 33.
Takeaway

Leading digits are really a pair of inequalities -- squeeze with them.

  • Turn the digits into an inequality
  • Pin down the numerator
  • Find the allowed denominators
  • Keep lowest terms and maximize
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