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Number Pyramid Problem: Find ★ + ♥ with Top 40 and Bottom 7, ★, 9
Mathematics
Primary School (Grade 5-6)
Question Content
In a number pyramid, each number in the middle and top rows is the sum of the two numbers below it. Given a pyramid with top 40, bottom row 7, ★, 9, and middle row (left) and ♥ (right), find the value of ★ + ♥. Options: A 25, B 33, C 45, D 57, E 65.
Correct Answer
33
Detailed Solution Steps
1
1. Recall the number pyramid rule: Each number in a row is the sum of the two numbers directly below it.
2
2. Let the bottom row be 7, ★, 9. Let the middle row left be \( L \) and right be \( \heartsuit \). The top (40) is \( L + \heartsuit \).
3
3. Express \( L \) and \( \heartsuit \) in terms of \( \bigstar \): \( L = 7 + \bigstar \) (sum of 7 and \( \bigstar \)), \( \heartsuit = \bigstar + 9 \) (sum of \( \bigstar \) and 9).
4
4. Substitute \( L \) and \( \heartsuit \) into the top row equation: \( (7 + \bigstar) + (\bigstar + 9) = 40 \).
5
5. Simplify: \( 7 + \bigstar + \bigstar + 9 = 40 \) → \( 2\bigstar + 16 = 40 \).
6
6. Solve for \( \bigstar \): \( 2\bigstar = 40 - 16 = 24 \) → \( \bigstar = 12 \).
7
7. Find \( \heartsuit \): \( \heartsuit = \bigstar + 9 = 12 + 9 = 21 \).
8
8. Calculate \( \bigstar + \heartsuit \): \( 12 + 21 = 33 \).
Knowledge Points Involved
1
Number Pyramid Rule
In a number pyramid, each number in a row (middle or top) is the sum of the two numbers directly below it. This requires applying addition and algebraic reasoning to solve for unknowns.
2
Algebraic Equation Solving
Setting up equations with unknowns (e.g., \( \bigstar \), \( \heartsuit \)) and solving using basic arithmetic operations (combining like terms, isolating variables).
3
Addition of Integers
Summing integers to find missing values in the pyramid, following the given rule of the pyramid’s structure.
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