Question 15 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.
Answer
View full question & answer→If the two numbers are x and y , then :
$\begin{array}{l}x \times y=(x+2) \times(y-4) \\x y=(x+2) y-(x+2) 4 \\x y=x y+2 y-(4 x+8) \\x y=x y+2 y-4 x-8 \\x y-x y=2 y-4 x-8 \\0=2 y-4 x-8 \\4 x+8=2 y \\2(2 x+4)=2 y \\y=2 x+4\end{array}$
Examples :
(i) $x=1, y=6 \rightarrow$ Product $=1 \times 6=6$
Check: $(1+2) \times(6-4)=3 \times 2=6$.
(ii) $x=2, y=8 \rightarrow$ Product $=16$
Check: $(2+2) \times(8-4)=4 \times 4=16$.
(iii) $x =5, y =14 \rightarrow$ Product $=5 \times 14=70$
Check: $(5+2) \times(14-4)=7 \times 10=70$.
Therefore, $(1,6),(2,8)$, and $(5,14)$ are three valid examples.
$\begin{array}{l}x \times y=(x+2) \times(y-4) \\x y=(x+2) y-(x+2) 4 \\x y=x y+2 y-(4 x+8) \\x y=x y+2 y-4 x-8 \\x y-x y=2 y-4 x-8 \\0=2 y-4 x-8 \\4 x+8=2 y \\2(2 x+4)=2 y \\y=2 x+4\end{array}$
Examples :
(i) $x=1, y=6 \rightarrow$ Product $=1 \times 6=6$
Check: $(1+2) \times(6-4)=3 \times 2=6$.
(ii) $x=2, y=8 \rightarrow$ Product $=16$
Check: $(2+2) \times(8-4)=4 \times 4=16$.
(iii) $x =5, y =14 \rightarrow$ Product $=5 \times 14=70$
Check: $(5+2) \times(14-4)=7 \times 10=70$.
Therefore, $(1,6),(2,8)$, and $(5,14)$ are three valid examples.