If 30% of a number is equal to \frac{1}{6} of the second number and the sum of these numbers is 140, then the value of larger number is :

Question

Accepted Answer

To solve the problem and justify why Option 4 is the correct answer, let's break down the given information and perform the necessary calculations.

We are given two conditions:
1. 30% of the first number (let's call it $ x $) is equal to $\frac{1}{6}$ of the second number (let's call it $ y $).
2. The sum of these two numbers is 140.

Let's translate these conditions into equations:

1. $ 0.3x = \frac{1}{6}y $
2. $ x + y = 140 $

From the first equation, we can express $ y $ in terms of $ x $:

$$ y = 6 \times 0.3x = 1.8x $$

Now, substitute $ y = 1.8x $ into the second equation:

$$ x + 1.8x = 140 $$

$$ 2.8x = 140 $$

Solve for $ x $:

$$ x = \frac{140}{2.8} = 50 $$

Now that we have $ x = 50 $, we can find $ y $:

$$ y = 1.8 \times 50 = 90 $$

Thus, the larger number is $ y = 90 $.

Now, let's explain why Option 4 is correct and why the other options are incorrect:

1. **Option 1 (110):** This cannot be correct because when we calculated $ y $, we found it to be 90, not 110. If $ y $ were 110, the sum $ x + y $ would exceed 140, which contradicts the given condition.

2. **Option 2 (50):** This is the value of $ x $, the smaller number, not the larger number. Therefore, it cannot be the correct answer for the larger number.

3. **Option 3 (60):** This value does not satisfy the equation $ y = 1.8x $ when $ x = 50 $. If $ y $ were 60, the sum $ x + y $ would not equal 140, which contradicts the given condition.

4. **Option 4 (90):** This is the correct answer because it satisfies both the condition $ y = 1.8x $ and the sum $ x + y = 140 $. The calculations confirm that when $ x = 50 $, $ y = 90 $, making 90 the larger number.

Therefore, Option 4 is the correct answer, as it aligns with the given conditions and the calculations performed.