Question 2 of 27intermediate🔧 ApplyShort Answer3 marks

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Correct Answer

Finding the condition: For product to remain unchanged: a×b=(a+2)(b−4)a \times b = (a+2)(b-4)

Expanding:

ab=ab−4a+2b−8ab = ab - 4a + 2b - 8

Simplifying:

−4a+2b−8=0-4a + 2b - 8 = 0 2b=4a+82b = 4a + 8 b=2a+4b = 2a + 4

Three examples satisfying b=2a+4b = 2a + 4:

(1, 6):

1×6=61 \times 6 = 6 (1+2)×(6−4)=3×2=6(1+2) \times (6-4) = 3 \times 2 = 6

∴ Product unchanged at 6.

(2, 8):

2×8=162 \times 8 = 16 (2+2)×(8−4)=4×4=16(2+2) \times (8-4) = 4 \times 4 = 16

∴ Product unchanged at 16.

(3, 10):

3×10=303 \times 10 = 30 (3+2)×(10−4)=5×6=30(3+2) \times (10-4) = 5 \times 6 = 30

∴ Product unchanged at 30.

Exercise: Figure it Out (Section 1) | Q: 3 | (Page Page 8)
For More Understanding

Explanation

The chapter discusses how products change when numbers are modified. Using Identity 1: (a+m)(b+n)=ab+mb+an+mn(a+m)(b+n) = ab + mb + an + mn. For product to remain unchanged when one increases by 2 and other decreases by 4, we need 2b−4a−8=02b - 4a - 8 = 0, giving b=2a+4b = 2a + 4.

Solution Steps

  1. Set up the condition: For product to remain unchanged, a×b=(a+2)(b−4)a \times b = (a+2)(b-4)

  2. Expand the right side: ab=ab−4a+2b−8ab = ab - 4a + 2b - 8

  3. Simplify the equation: −4a+2b−8=0-4a + 2b - 8 = 0

  4. Solve for the relationship: 2b=4a+82b = 4a + 8, giving b=2a+4b = 2a + 4

  5. Find pairs where b=2a+4b = 2a + 4: (1,6), (2,8), (3,10)

  6. Verify each pair shows unchanged product

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