X is 40% more efficient than Y and Z takes `33(1)/(3)%` less time than X to complete a work. If all three together can complete a work in `6(2)/(9)` days, then find X & Z can complete the two times of work in how many days?

Options

A. A)8 days
B. B)16 days
C. C)18 days
D. D)32 days

Answer

To solve the problem step by step, we will follow the reasoning and calculations as outlined in the video transcript.

### Step 1: Determine the Efficiency Ratio of X and Y
X is 40% more efficient than Y. If we assume Y's efficiency is 5 units, then:
- X's efficiency = Y's efficiency + 40% of Y's efficiency
- X's efficiency = 5 + (40/100) * 5 = 5 + 2 = 7 units

Thus, the efficiency ratio of X to Y is:
- X : Y = 7 : 5

**Hint:** To find the efficiency of X when given a percentage increase, convert the percentage to a fraction and apply it to the base efficiency.

### Step 2: Determine the Efficiency Ratio of Z
Z takes 33⅓% less time than X. This means Z is more efficient than X.
- 33⅓% can be expressed as 1/3.
If X takes 3 units of time, then Z takes:
- Z's time = 3 - (1/3 * 3) = 3 - 1 = 2 units

Since efficiency is inversely proportional to time:
- Efficiency of Z = 1 / (time taken by Z) = 3/2 units

Thus, the efficiency ratio of X to Z is:
- X : Z = 2 : 3

**Hint:** When time decreases by a certain percentage, the efficiency increases by the same proportion. Use the inverse relationship between time and efficiency.

### Step 3: Combine the Efficiency Ratios
Now we have:
- X : Y = 7 : 5
- X : Z = 2 : 3

To combine these ratios, we need a common term for X. We can express the ratios as:
- X : Y : Z = 14 : 10 : 21

**Hint:** To combine ratios, find a common term (in this case, X) and adjust the other parts of the ratio accordingly.

### Step 4: Calculate Total Work Done
We know that all three together can complete the work in \(6 \frac{2}{9}\) days. First, convert this to an improper fraction:
- \(6 \frac{2}{9} = \frac{56}{9}\) days.

Total efficiency of X, Y, and Z:
- Total efficiency = 14 + 10 + 21 = 45 units.

Total work done can be calculated as:
- Total work = Total efficiency × Time = 45 × \( \frac{56}{9} \)
- Total work = \( \frac{2520}{9} = 280 \) units.

**Hint:** To find total work, multiply the total efficiency by the time taken to complete the work.

### Step 5: Calculate Time Taken by X and Z to Complete Double the Work
If X and Z work together to complete double the work:
- Double work = 280 × 2 = 560 units.

Efficiency of X and Z together:
- Efficiency of X and Z = 14 + 21 = 35 units.

Time taken by X and Z to complete double the work:
- Time = Total work / Combined efficiency = \( \frac{560}{35} \)
- Time = 16 days.

**Hint:** When calculating time for a given amount of work, divide the total work by the combined efficiency of the workers.

### Final Answer
X and Z can complete double the work in **16 days**.