Problem 24 · 2024 Math Kangaroo
Stretch
Algebra & Patterns
place-valuesubstitution
I have a four-digit number \(N=\overline{pqrs}\). If I place a decimal point between the digits q and r, I obtain the number \(\overline{pq}.\overline{rs}\). This is exactly the average of the two numbers \(\overline{pq}\) and \(\overline{rs}\). What is the sum of the digits of N?
Show answer
Answer: B — 18
Show hints
Hint 1 of 2
Turning pq.rs into an average gives an equation linking the two-digit blocks pq and rs.
Still stuck? Show hint 2 →
Hint 2 of 2
From pq + rs/100 = (pq + rs)/2 you get 50·pq = 49·rs, which pins down pq and rs.
Show solution
Approach: translate the average condition into an equation
- The number with the decimal is pq + rs/100, and it equals (pq + rs)/2.
- Clearing fractions: 100·pq + rs = 50·pq + 50·rs, so 50·pq = 49·rs.
- The two-digit solution is pq = 49, rs = 50, so N = 4950 and the digit sum is 4+9+5+0 = 18.
Mark:
· log in to save