X is `35%` less than Y, Z is `40%` more than the sum of X and Y . By what percent is Z more than 2 times X (correct to one decimal place) ?

Options

A. `75.8%`
B. `78.1%`
C. `75.4%`
D. `77.7%`

Answer

To solve the problem step by step, let's define the variables and work through the calculations:

1. **Define the relationship between X and Y:**
- We know that X is 35% less than Y.
- This can be expressed as:
\[ X = Y - 0.35Y = 0.65Y \]

2. **Calculate the sum of X and Y:**
- The sum of X and Y is:
\[ X + Y = 0.65Y + Y = 1.65Y \]

3. **Calculate Z:**
- Z is 40% more than the sum of X and Y.
- Therefore, we can express Z as:
\[ Z = 1.40 \times (X + Y) = 1.40 \times 1.65Y \]
- Now, calculate Z:
\[ Z = 1.40 \times 1.65Y = 2.31Y \]

4. **Calculate 2 times X:**
- We know that:
\[ 2X = 2 \times 0.65Y = 1.30Y \]

5. **Find the difference between Z and 2X:**
- Now, we need to find out how much Z is more than 2 times X:
\[ Z - 2X = 2.31Y - 1.30Y = 1.01Y \]

6. **Calculate the percentage increase of Z over 2X:**
- The percentage increase can be calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{Z - 2X}{2X} \right) \times 100 \]
- Substitute the values:
\[ \text{Percentage Increase} = \left( \frac{1.01Y}{1.30Y} \right) \times 100 \]
- The Y cancels out:
\[ \text{Percentage Increase} = \left( \frac{1.01}{1.30} \right) \times 100 \]
- Calculate the fraction:
\[ \frac{1.01}{1.30} \approx 0.7769 \]
- Now multiply by 100:
\[ \text{Percentage Increase} \approx 77.69\% \]

7. **Round to one decimal place:**
- Therefore, the final answer is:
\[ 77.7\% \]

### Summary of Steps:
1. Define X in terms of Y.
2. Calculate the sum of X and Y.
3. Calculate Z based on the sum.
4. Calculate 2 times X.
5. Find the difference between Z and 2X.
6. Calculate the percentage increase.
7. Round to one decimal place.