Adding Fractions by Multiplying

Algebra Level 3

Daniel Z. is an elementary schooler trying to solve $\dfrac{5}{2}+\dfrac{5}{3}$ . However, he mixes up fraction addition with fraction multiplication and gets $\dfrac{5}{2}+\dfrac{5}{3} = \dfrac{5 \times 5}{2 \times 3} = \dfrac{25}{6}$ .

When Daniel's teacher graded the paper, she realized that even though he solved the problem incorrectly, he got the right answer! As a math enthusiast, she tried to generalize this:

Let $a$ , $b$ , $c$ , and $d$ be non-zero real numbers satisfying $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ac}{bd}$ . Which of the following statements must be true?

ad - bc = \pm a \text{ }

\text{ }ad - bc = \pm c \text{ }

a-b \text{ }

and

\text{ } c-d \text{ }

differ by

1

\frac{b}{a} + \frac{d}{c} = 1

4bd + 1 = ac

2 solutions

Zain Majumder
Dec 5, 2018

$\frac{a}{b} + \frac{c}{d} = \frac{ac}{bd}$

$\frac{a{\color{#D61F06}d}}{b{\color{#D61F06}d}} + \frac{{\color{#D61F06}b}c}{{\color{#D61F06}b}d} = \frac{ac}{bd}$

${\color{#3D99F6}\frac{bd}{ac}} \times\frac{a{\color{#D61F06}d}}{b{\color{#D61F06}d}} + {\color{#3D99F6}\frac{bd}{ac}}\times\frac{{\color{#D61F06}b}c}{{\color{#D61F06}b}d} = {\color{#3D99F6}\frac{bd}{ac}}\times\frac{ac}{bd}$

$\boxed{\frac{b}{a} + \frac{d}{c} = 1}$

A counterexample to the other three choices is $\frac{2}{1} + \frac{2}{1} = \frac{2 \times 2}{1 \times 1} = 4$ .

Chew-Seong Cheong
Dec 6, 2018

$\begin{aligned} \frac ab + \frac cd & = \frac {ac}{bd} \\ \frac {ad+bc}{bd} & = \frac {ac}{bd} & \small \color{#3D99F6} \text{Multiply both sides by }bd. \\ \implies ad + bc & = ac & \small \color{#3D99F6} \text{Divide both sides by }ac. \\ \frac dc + \frac ba & = 1 \end{aligned}$

Therefore, the answer is $\boxed{\dfrac ba + \dfrac dc = 1}$ .

Adding Fractions by Multiplying

2 solutions

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