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?​

Answer 1

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

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Answer 2

Hence, the answer is 52.

Given: Sums of every four pair out of five.

To Find: The middle number that the 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.

Hence, the correct answer is 52.