Question :- A boy IMAGINED five positive integer numbers and told his friend that the sums of four out of five numbers he imagined are 186, 203, 214, 228, and 233. What is the middle number that boy imagined ?

Solution :-

Let us assume that, five positive integer numbers are a, b , c, d and e .

so, we have,

→ a + b + c + d = 186 ---------------- Eqn.(1)

→ b + c + d + e = 203 ---------------- Eqn.(2)

→ c + d + e + a = 214 ---------------- Eqn.(3)

→ d + e + a + b = 228 ---------------- Eqn.(4)

→ e + a + b + c = 233 ---------------- Eqn.(5)

ADDING all 5 Equations we get,

→ (a + b + c + d) + (b + c + d + e) + (c + d + e + a) + (d + e + a + b) + (e + a + b + c) = 186 + 203 + 214 + 228 + 233

→ 4a + 4b + 4c + 4d + 4e = 1064

→ 4(a + b + c + d + e) = 1064

→ (a + b + c + d + e) = 266 ---------------- Eqn.(6)

therefore,

• a = Eqn.(6) - Eqn.(2) = 266 - 203 = 63 .
• b = Eqn.(6) - Eqn.(3) = 266 - 214 = 52 .
• c = Eqn.(6) - Eqn.(4) = 266 - 228 = 38 .
• d = Eqn.(6) - Eqn.(5) = 266 - 233 = 33 .
• e = Eqn.(6) - Eqn.(1) = 266 - 186 = 80 .

as we can see ,

→ 33 < 38 < 52 < 63 < 80 .

Hence, the middle number that boy imagined (in increasing order) is 52.

Learn more :-

If the sum of 3 natural numbers a,b, and c is 99 and a has 3 divisors then what will be the minimum value of b+c.

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Target Olympiad

1. Find the values of A and B if the given number 7A5798B8 is divisible by 33.

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