the sum of two numbers is 688. if their numerator is 7, find each number.

The problem statement seems a bit unclear because it mentions "numerator" in a context where it doesn't seem directly relevant. However, I will proceed with the assumption that the sum of two numbers is given, and we need to find these numbers under the condition that their ratio is 7:1. 

Let's break down the problem as follows:

1. Let the two numbers be \( x \) and \( y \).
2. According to the problem, their sum is 688:
   \[
   x + y = 688
   \]
3. We are also given that the ratio of these numbers is 7:1, which can be written as:
   \[
   \frac{x}{y} = 7
   \]
   This implies that \( x = 7y \).

### Step-by-Step Solution:

1. Substitute \( x = 7y \) into the equation \( x + y = 688 \):
   \[
   7y + y = 688
   \]
   \[
   8y = 688
   \]

2. Solve for \( y \):
   \[
   y = \frac{688}{8} = 86
   \]

3. Now find \( x \) using \( x = 7y \):
   \[
   x = 7 \times 86 = 602
   \]

### Conclusion:

The two numbers are:
- \( x = 602 \)
- \( y = 86 \)

### Verification:

- Sum: \( 602 + 86 = 688 \)
- Ratio: \( \frac{602}{86} = 7 \)

Both conditions are satisfied.

Would you like any further details or have any questions?

### Related Questions:

1. What would the numbers be if their ratio were 3:2 instead?
2. How would you find the numbers if their sum were 1000 and their ratio was 4:1?
3. What is the process to solve for numbers if you only know their product and sum?
4. How can we generalize the solution for any given ratio \(a:b\) and sum \(S\)?
5. How does this problem change if we add the condition that the numbers must be integers?

### Tip:

When dealing with ratios and sums, set up equations using variables to represent the unknown quantities, and use algebraic substitution to solve them step-by-step.

Learn how to find two numbers when their sum is 688 and their ratio is 7:1. Follow a detailed step-by-step solution using algebraic equations and ratio concepts. Suitable for Grades 7-9.

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