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Algebra Play - Number Pyramids

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Number Pyramid is a visual math puzzle where the value in each cell is the sum of the two cells directly below it.

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If the two adjacent cells in a lower row are aa and bb, the cell directly above them will contain the expression a+ba + b.

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Algebraic Number Pyramids use variables like xx and yy to represent unknown values in the blocks.

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To solve for an unknown variable in a pyramid, we construct a linear equation based on the addition rule and solve for the variable.

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The 'Apex' is the topmost cell of the pyramid, which represents the final sum of the expressions built from the base upward.

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Subtraction (inverse operation) can be used to find a missing value in a lower row if the upper cell and one lower cell are known: a=(a+b)−ba = (a+b) - b.

📐Formulae

C=A+BC = A + B

A=C−BA = C - B

Apex of a 3-layer pyramid with base x,y,z=(x+y)+(y+z)=x+2y+z\text{Apex of a 3-layer pyramid with base } x, y, z = (x + y) + (y + z) = x + 2y + z

Apex of a 4-layer pyramid with base a,b,c,d=a+3b+3c+d\text{Apex of a 4-layer pyramid with base } a, b, c, d = a + 3b + 3c + d

💡Examples

Problem 1:

In a 3-tier number pyramid, the bottom row has the values xx, 77, and 1212 (from left to right). If the value at the top of the pyramid is 4545, find the value of xx.

Solution:

  1. Find the values for the middle row:
  • Left cell: x+7x + 7
  • Right cell: 7+12=197 + 12 = 19
  1. Find the expression for the top cell (Apex):
  • (x+7)+19=x+26(x + 7) + 19 = x + 26
  1. Set up the equation:
  • x+26=45x + 26 = 45
  • x=45−26x = 45 - 26
  • x=19x = 19

Explanation:

We use the rule that each block is the sum of the two below it to work our way up to the apex. Once we have an expression for the apex in terms of xx, we equate it to the given numerical value.

Problem 2:

Complete the algebraic pyramid where the base row is 2a2a, 3a3a, and 4a4a. What is the expression in the top box?

Solution:

  1. Calculate the middle layer:
  • Left middle: 2a+3a=5a2a + 3a = 5a
  • Right middle: 3a+4a=7a3a + 4a = 7a
  1. Calculate the top layer:
  • Top: 5a+7a=12a5a + 7a = 12a
  1. Final result: 12a12a

Explanation:

By adding adjacent terms, we combine like terms. 2a2a and 3a3a are like terms, so their sum is (2+3)a=5a(2+3)a = 5a. Similarly, 5a5a and 7a7a sum to 12a12a.

Problem 3:

Determine the value of yy if the top box of a 2-tier pyramid is 15y+1015y + 10 and the bottom-left box is 5y+25y + 2.

Solution:

Let the bottom-right box be BB.

  1. According to the rule: (5y+2)+B=15y+10(5y + 2) + B = 15y + 10
  2. Solve for BB using subtraction: (15y+10)−(5y+2)10y+8\begin{array}{r} (15y + 10) \\ - (5y + 2) \\ \hline 10y + 8 \end{array}
  3. Therefore, the missing box is 10y+810y + 8.

Explanation:

To find a missing addend in the bottom row, subtract the known bottom value from the top value. Subtract coefficients of yy and constants separately.