80% of a number is equal to the $\frac{4}{5}$th of the other number. What is the ratio between the first number and the second number respectively? [United India Insurance Co. Ltd. (UIICL) Assistant (Online) Exam, 2015]

### Concept & Equation Setup Translate the literal text into a mathematical equation. "80%" is equivalent to the fraction $\frac{80}{100}$ which simplifies to $\frac{4}{5}$. ### Step-by-Step Solution 1. Let the first number be $x$ and the second number be $y$. 2. The problem states that 80% of the first number ($x$) equals $\frac{4}{5}$ of the second number ($y$). $$ 80\% \text{ of } x = \frac{4}{5} \text{ of } y $$ 3. Convert the percentage to a fraction: $$ \frac{80}{100} \times x = \frac{4}{5} \times y $$ 4. Simplify the fraction on the left side: $$ \frac{4}{5}x = \frac{4}{5}y $$ 5. Divide both sides by $\frac{4}{5}$: $$ x = y $$ 6. To find the ratio between the first and second number ($x : y$): $$ \frac{x}{y} = \frac{1}{1} $$ $$ \text{Ratio} = 1 : 1 $$ 7. Look at the options: (a) 3 : 4, (b) 3 : 5, (c) 5 : 3. The ratio 1 : 1 is not present. 8. Hence, the correct choice is "None of these". ### Exam Strategy & Shortcut Recognize standard fraction conversions instantly. 80% is $\frac{4}{5}$. The question is literally stating "$\frac{4}{5}$ of number A equals $\frac{4}{5}$ of number B". This immediately implies A = B, meaning a 1 : 1 ratio. It requires zero written calculation. ### Common Pitfall A common pitfall is getting a 1:1 ratio, panicking because it's not a numerical option, assuming you made a calculation error, and then randomly guessing one of the other options instead of confidently choosing "None of these". ### Final Answer Therefore, the correct answer is **None of these**.