Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

12. Observe the following equation:
0×0=0−0,1×12=1−12,⋯.  0 \times 0=0-0,1 \times \frac{1}{2}=1-\frac{1}{2}, \cdots \text {. }

According to the pattern reflected in the equation, write two more equations that satisfy this pattern. \qquad

A number or a short expression. Spacing and $ signs are ignored.

Solution

12. 2×23=2−23;3×34=3−342 \times \frac{2}{3}=2-\frac{2}{3} ; 3 \times \frac{3}{4}=3-\frac{3}{4}.

If we use xx and yy to represent these two numbers, the pattern reflected in the equation can be expressed as xy=x−yxy=x-y. Therefore, we have
y=x1+x. y=\frac{x}{1+x} .

Taking x=2x=2, then y=23y=\frac{2}{3}; taking x=3x=3, then y=34y=\frac{3}{4}.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.