How can I divide 1400 into three parts where the first part is 23 times the second part, and the second part has a ratio of 4:5 with the third part?

To solve the problem of dividing 1400 into three parts based on the given conditions, we can start by defining our three parts.

Let’s denote the three parts as:

According to the problem:

The ratio between the second part (y) and the third part (z) is 4:5, which can be expressed as:

Now, let’s substitute the expressions we have into the equation for the sum:

Using x = 23y in the sum equation, we get:

23y + y + z = 1400

This simplifies to:

24y + z = 1400

From the ratio we derived (5y = 4z), we can express z in terms of y:

z = (5/4)y

Now substituting this expression for z back into our sum equation:

24y + (5/4)y = 1400

To eliminate the fraction, multiply the entire equation by 4:

4 * 24y + 5y = 5600

Which simplifies to:

96y + 5y = 5600

101y = 5600

Now, solving for y gives:

y = 5600 / 101 ≈ 55.44

Now that we have y, we can find z using z = (5/4)y:

z = (5/4) * 55.44 ≈ 69.30

Finally, we can calculate x using x = 23y:

x = 23 * 55.44 ≈ 1275.26

The three parts approximate values can be summarized as:

To make sure these values add up correctly:

x + y + z ≈ 1275.26 + 55.44 + 69.30 ≈ 1400

This confirms our calculations were correct. Therefore, the approximate parts of 1400 are: