Blocks...

Algebra Level 2

What is the number in the green block? (Given that the number in a block is the sum of the numbers in the two blocks below it.)


This is part of the series: " It's easy, believe me! "


The answer is 15.

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2 solutions

Zico Quintina
Jun 26, 2018

We call the values in the three empty blocks in the bottom row α , β \alpha, \beta and γ \gamma . Then we add our way up the pyramid as shown below:

Then from the pairs of blocks beneath the three red blocks in the top and middle rows, we get the equations

120 + 2 α + 4 β + 2 γ = 240 α + 2 β + γ = 60 ( 1 ) 12 + 2 α + β = 67 2 α + β = 55 ( 2 ) β + 2 γ + 18 = 53 β + 2 γ = 35 ( 3 ) α + β + γ = 45 ( 4 ) [ Half of sum ( 2 ) + ( 3 ) ] β = 15 [ ( 1 ) ( 4 ) ] \begin{array}{rlcrlccl} 120 + 2\alpha + 4\beta + 2\gamma &= \ \ 240 \ \ & \implies &\ \ \alpha + 2\beta + \gamma &= \ \ 60 & \small (1) \\ 12 + 2\alpha + \beta &= \ \ 67 & \implies & 2\alpha + \beta &= \ \ 55 & \small (2) \\ \beta + 2\gamma + 18 &= \ \ 53 & \implies & \beta + 2\gamma &= \ \ 35 & \small (3) \\ \\ & & & \alpha + \beta + \gamma &= \ \ 45 & \small (4) & &\small \text{[ Half of sum } (2) + (3) \ ] \\ \\ & & & \beta &= \ \ \boxed{15} & & &\small [ \ (1) - (4) \ ] \end{array}

In figure 1, let x x be the unknown number on the second row leftmost, we observed that a = x 67 a=x-67 and a = 240 x 53 = 187 x a=240-x-53=187-x . Then, equate the two equations

a = a a=a

x 67 = 187 x x-67=187-x

2 x = 254 2x=254

x = 127 x=127

It follows that 240 x = 240 127 = 113 240-x=240-127=113 and a = x 67 = 127 67 = 60 a=x-67=127-67=60

Fill up the other boxes as shown in figure 2.

In figure 3, let y y be the unknown number on the fourth row leftmost.

In figure 4, let m m be the number on the green box. We observed that m = 67 y ( y 12 ) = 67 y y + 12 = 79 2 y m=67-y-(y-12)=67-y-y+12=79-2y and m = y 7 ( 42 y ) = y 7 42 + y = 2 y 49 m=y-7-(42-y)=y-7-42+y=2y-49 . Equate the two equations

m = m m=m

79 2 y = 2 y 49 79-2y=2y-49

79 + 49 = 4 y 79+49=4y

y = 32 y=32

It follows that,

m = 79 2 ( 32 ) = 15 m=79-2(32)=\boxed{15}

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