A number is 20 percent smaller than a second number. However, the second number now increases by 20 percent with no change in the first number. What is the ratio of the two numbers now?
2 : 3
Understanding the Problem: Ratio and Percentage Changes
The question asks for the ratio of two numbers after one of them undergoes a percentage increase, while the other number is initially defined by a percentage decrease relative to the first. We need to track the values of both numbers through the described changes.
Setting Up the Numbers
Let's represent the two numbers. It's often helpful to start with the number that the other is based on. The first number is 20 percent smaller than the second number. This means the second number is the reference point for the initial relationship.
Let the initial value of the second number be $S$.
The first number is 20 percent smaller than the second number.
20 percent of the second number is $\frac{20}{100} \times S = 0.20S$.
The first number (let's call it $F$) is the second number minus 20 percent of the second number:
$F = S - 0.20S = (1 - 0.20)S = 0.80S$.
So, initially, the first number is $0.8S$ and the second number is $S$.
Applying the Percentage Increase
The problem states that the second number now increases by 20 percent. The first number does not change.
The increase in the second number is 20 percent of its current value ($S$).
Increase amount = $20\%$ of $S = 0.20S$.
The new value of the second number (let's call it $S_{new}$) is the original value plus the increase:
$S_{new} = S + 0.20S = (1 + 0.20)S = 1.20S$.
The first number remains unchanged, so its value is still $F = 0.8S$.
Calculating the Final Ratio
We need to find the ratio of the two numbers now. The question asks for the ratio of the first number to the second number.
Ratio = $\frac{\text{First Number}}{\text{New Second Number}}$
Ratio = $\frac{0.8S}{1.2S}$
To simplify the ratio, we can cancel out the variable $S$ (assuming $S \neq 0$, which must be true for percentage changes to be meaningful in this context) and simplify the decimal fraction:
Ratio = $\frac{0.8}{1.2} = \frac{8}{12}$
Now, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4:
Ratio = $\frac{8 \div 4}{12 \div 4} = \frac{2}{3}$
The ratio of the two numbers now is 2 : 3.
Summary of the Calculation Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Define initial second number | $S$ |
| 2 | Calculate initial first number (20% smaller than $S$) | $F = S - 0.20S = 0.8S$ |
| 3 | Calculate new second number (20% increase on $S$) | $S_{new} = S + 0.20S = 1.2S$ |
| 4 | First number remains unchanged | $F = 0.8S$ |
| 5 | Calculate the ratio of the first number to the new second number | Ratio = $\frac{F}{S_{new}} = \frac{0.8S}{1.2S} = \frac{0.8}{1.2}$ |
| 6 | Simplify the ratio | $\frac{0.8}{1.2} = \frac{8}{12} = \frac{2}{3}$ |
The final ratio of the two numbers is 2 : 3.
Revision Table: Percentage and Ratio Concepts
| Concept | Explanation | Example |
|---|---|---|
| Percentage Decrease | Reducing a quantity by a percentage. If a quantity $X$ decreases by $P\%$, the new quantity is $X - \frac{P}{100}X = X(1 - \frac{P}{100})$. | 100 decreased by 20% is $100 \times (1 - 0.20) = 100 \times 0.80 = 80$. |
| Percentage Increase | Increasing a quantity by a percentage. If a quantity $X$ increases by $P\%$, the new quantity is $X + \frac{P}{100}X = X(1 + \frac{P}{100})$. | 100 increased by 20% is $100 \times (1 + 0.20) = 100 \times 1.20 = 120$. |
| Ratio | A comparison of two quantities, often expressed as a fraction or using a colon. The ratio of $a$ to $b$ is $\frac{a}{b}$ or $a:b$. | The ratio of 8 to 12 is $\frac{8}{12}$ or $8:12$, which simplifies to $\frac{2}{3}$ or $2:3$. |
Additional Information: Working with Percentages
When dealing with percentage changes, it's crucial to identify the base amount on which the percentage is calculated. In this problem, the first number's initial value is based on the initial value of the second number. The increase in the second number is also based on its original value.
A common mistake is to assume that a 20% decrease followed by a 20% increase on the same *original* number brings you back to the start, but here the percentage changes are applied to different bases or sequentially to the result of the previous step (although in this specific problem, the first number calculation is independent of the second number's change).
Using decimal multipliers (like 0.80 for a 20% decrease and 1.20 for a 20% increase) can simplify calculations involving percentages.
Important Questions from Simple Ratios
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct ?
A bag contains Rs. 420 in the form of 5 rupee, 2 rupee, and 1 rupee coins. The number of coins of 5 rupee, 2 rupee, and 1 rupee is in the ratio of 2 : 3 : 5. What is the total number of coins in the bag?
If P/Q = 8/9 and Q - P = 12, then what is the value of P?
A child gets 1 random coin out of Re 1, Rs 2, and Rs 5 coins daily. He starts on 1st April and at the end of the month, he has coins in the ratio 5 ∶ 4 ∶ 6. Find the value of the coins with him
Find the proportion equivalent to M : N : O : P if M : N = 8 : 15, N : O = 5 : 9 and O : P = 18 : 19 ?
