Understanding the Problem: Dividing a Number into Parts

The question asks us to divide a total number, which is 800, into two distinct parts. We are given a specific relationship between these two parts: one part is \(\frac{9}{7}\) times the other part. Our goal is to find the value of the larger of these two parts.

Let's represent the two unknown parts using variables. Suppose the two parts are \(x\) and \(y\).

Setting Up Equations from the Problem Description

From the problem statement, we can form two equations:

1. The sum of the two parts is 800. This gives us the equation:

\[x + y = 800\]

2. One part is \(\frac{9}{7}\) times the other. Let's assume that \(y\) is \(\frac{9}{7}\) times \(x\). This gives us the equation:

\[y = \frac{9}{7}x\]

Now we have a system of two linear equations with two variables, \(x\) and \(y\). We can solve this system to find the values of \(x\) and \(y\).

Solving the System of Equations

We can use the substitution method to solve this system. Since we already have \(y\) expressed in terms of \(x\) from the second equation, we can substitute the expression for \(y\) into the first equation.

Substitute \(y = \frac{9}{7}x\) into the equation \(x + y = 800\):

\[x + \left(\frac{9}{7}x\right) = 800\]

To solve for \(x\), we need to combine the terms involving \(x\). We can rewrite \(x\) as \(\frac{7}{7}x\) so that it has the same denominator as the other term:

\[\frac{7}{7}x + \frac{9}{7}x = 800\]

Now, add the fractions:

\[\frac{7x + 9x}{7} = 800\] \[\frac{16x}{7} = 800\]

To isolate \(x\), multiply both sides of the equation by 7:

\[16x = 800 \times 7\] \[16x = 5600\]

Now, divide both sides by 16 to find the value of \(x\):

\[x = \frac{5600}{16}\]

Let's perform the division:

\[x = 350\]

So, one part is 350.

Finding the Second Part

Now that we have the value of \(x\), we can find the value of \(y\) using the equation \(y = \frac{9}{7}x\):

\[y = \frac{9}{7} \times 350\]

We can simplify this calculation by dividing 350 by 7 first:

\[y = 9 \times \left(\frac{350}{7}\right)\] \[y = 9 \times 50\] \[y = 450\]

So, the second part is 450.

Identifying the Larger Part

The two parts are 350 and 450. We need to find the larger part. Comparing the two values:

• Part 1 (\(x\)) = 350
• Part 2 (\(y\)) = 450

Clearly, 450 is greater than 350.

We can also verify that 450 is indeed \(\frac{9}{7}\) times 350:

\[\frac{9}{7} \times 350 = 9 \times \frac{350}{7} = 9 \times 50 = 450\]

This confirms our values for \(x\) and \(y\) are correct and satisfy the condition.

Conclusion

The number 800 is divided into two parts, 350 and 450. The larger part is 450.

Step-by-Step Summary

• Assign variables (\(x\), \(y\)) to the two parts.
• Write the equation for the sum: \(x + y = 800\).
• Write the equation for the ratio: \(y = \frac{9}{7}x\) (or \(x = \frac{9}{7}y\)).
• Substitute the ratio equation into the sum equation.
• Solve the resulting linear equation for one variable (\(x\)).
• Substitute the value of the first variable back into the ratio equation to find the second variable (\(y\)).
• Compare the two values to find the larger part.

Revision Table: Key Concepts

Additional Information: Ratios and Proportions

This problem involves the concept of ratios and proportions. When a number is divided into two parts in a certain ratio, say \(a:b\), it means the parts can be represented as \(ak\) and \(bk\) for some constant \(k\). The sum of the parts is \(ak + bk = (a+b)k\). In this problem, one part is \(\frac{9}{7}\) times the other. If one part is \(x\), the other is \(\frac{9}{7}x\). This implies a ratio of \(1 : \frac{9}{7}\), or multiplying by 7, a ratio of \(7:9\).

So, the number 800 is divided in the ratio \(7:9\). The total number of "ratio units" is \(7 + 9 = 16\).

Each ratio unit represents \(\frac{800}{16} = 50\).

The two parts are:

• First part (7 units): \(7 \times 50 = 350\).
• Second part (9 units): \(9 \times 50 = 450\).

This ratio method confirms our earlier result and provides an alternative way to solve such problems involving dividing a quantity in a given ratio.

The larger part is indeed 450.