# A number is divided into three parts such that three times the first part, six times the second part and eight times the third part are equal. If the first part is Rs. 1600, then what is the third part ?

एक संख्या को तीन भागों में इस प्रकार विभाजित किया जाता है की पहला भाग का तीन गुना, दूसरे भाग का छह गुना और तीसरे भाग का आठ गुना बराबर है | यदि पहला भाग Rs 1600 है, तो तीसरी भाग कितना है?

Rs. 450

Rs. 900

Rs. 600

Rs. 750

To solve the problem step by step, we start by defining the three parts of the number as follows:

Let:

- First part = x

- Second part = y

- Third part = z

According to the problem, we have the following relationships:

1. 3×first part=6×second part=8×third part

This means:

- 3x=6y

- 6y=8z

**Step 1: Establish the ratios**

From the first equation 3x=6y, we can simplify it:

xy=63=2⟹x:y=2:1

From the second equation 6y=8z, we can simplify it:

yz=86=43⟹y:z=4:3

**Step 2: Combine the ratios**

Now we have:

- x:y=2:1

- y:z=4:3

To combine these ratios, we need to express them with a common term for y. We can set y=1 in the first ratio and y=4 in the second ratio.

To do this, we multiply the first ratio by 4:

x:y=2×4:1×4=8:4

Now we can combine:

- x:y:z=8:4:3

**Step 3: Assign values based on the first part**

We know from the problem that the first part x=1600. This corresponds to the 8 parts in our ratio.

**Step 4: Calculate the value of one part**

To find the value of one part, we calculate:

1 part=16008=200

**Step 5: Find the third part**

The third part z corresponds to 3 parts in our ratio:

z=3 parts=3×200=600

**Conclusion**

Thus, the value of the third part is Rs. 600.

- First part = x

- Second part = y

- Third part = z

1. 3×first part=6×second part=8×third part

- 3x=6y

- 6y=8z

**Step 1: Establish the ratios**

From the first equation 3x=6y, we can simplify it:

xy=63=2⟹x:y=2:1

yz=86=43⟹y:z=4:3

**Step 2: Combine the ratios**

Now we have:

- x:y=2:1

- y:z=4:3

x:y=2×4:1×4=8:4

- x:y:z=8:4:3

**Step 3: Assign values based on the first part**

We know from the problem that the first part x=1600. This corresponds to the 8 parts in our ratio.

**Step 4: Calculate the value of one part**

To find the value of one part, we calculate:

1 part=16008=200

**Step 5: Find the third part**

The third part z corresponds to 3 parts in our ratio:

z=3 parts=3×200=600

**Conclusion**

Thus, the value of the third part is Rs. 600.

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