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Solve Number Pyramid Problem to Find ★ + ♥
Mathematics
Grade 5 (Primary School) or Junior High School Grade 7
Question Content
In a number pyramid, each number in the middle and top rows is the sum of the two numbers below it. The top number of a pyramid is 40, and the bottom row is 7, ★, 9. The middle row has a missing number (left) and ♥ (right). Find the value of ★ + ♥. Options: A 25, B 33, C 45, D 57, E 65.
Correct Answer
33 (Option B)
Detailed Solution Steps
1
1. Recall the number pyramid rule: Each number in the middle/top row is the sum of the two numbers directly below it.
2
2. Define variables: Let the bottom - middle number be ★, the left middle number be \( A \), and the right middle number be ♥.
3
3. Express middle - row numbers in terms of ★: \( A = 7 + ★ \) (since \( A \) is the sum of 7 and ★), and \( ♥ = ★ + 9 \) (since ♥ is the sum of ★ and 9).
4
4. Express the top number (40) in terms of ★: The top number is the sum of the two middle - row numbers, so \( 40 = A + ♥=(7 + ★)+(★ + 9) \).
5
5. Solve for ★: Simplify the equation \( 40=(7 + ★)+(★ + 9) \) to \( 40 = 16+2★ \). Subtract 16 from both sides: \( 2★ = 40 - 16 = 24 \). Then divide by 2: \( ★ = 12 \).
6
6. Solve for ♥: Substitute \( ★ = 12 \) into \( ♥ = ★ + 9 \), so \( ♥ = 12 + 9 = 21 \).
7
7. Calculate \( ★ + ♥ \): \( 12+21 = 33 \).
Knowledge Points Involved
1
Number Pyramid Rule
In a number pyramid, each number in the non - bottom rows is equal to the sum of the two numbers directly below it. This rule is the key to establishing equations to solve for unknown numbers in the pyramid.
2
Algebraic Equation Solving
By defining unknowns (like ★, \( A \), ♥ in this problem) and translating the relationship between numbers (from the number pyramid rule) into algebraic equations, we can solve for the unknown values. This involves simplifying equations, combining like terms, and using inverse operations (e.g., subtraction, division) to isolate the unknown.
3
Arithmetic Operations (Addition and Subtraction)
Basic arithmetic operations are used to combine terms, simplify equations, and calculate the final result. For example, combining \( 7 + 9 \) to get 16, and calculating \( 40 - 16 \) to find the value related to the unknown multiplier of ★.
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