Solving the Number Problem

Let the larger number be represented by $L$ and the smaller number by $S$. We need to translate the given information into mathematical equations.

Formulating the Equations

Condition 1: When 6 times the larger number ($6L$) is divided by the smaller number ($S$), the quotient is 7 and the remainder is 7. Using the division algorithm ($Dividend = Divisor \times Quotient + Remainder$), we get:

$6L = S \times 7 + 7$

$6L = 7S + 7 \quad (Equation\ 1)$

Condition 2: When 9 times the smaller number ($9S$) is divided by the larger number ($L$), the quotient is 7 and the remainder is 21. This gives us:

$9S = L \times 7 + 21$

$9S = 7L + 21 \quad (Equation\ 2)$

Step-by-Step Solution

We now have a system of two linear equations with two variables. We can solve this system using substitution.

1. Express $L$ from Equation 1: From $6L = 7S + 7$, we can isolate $L$: $L = \frac{7S + 7}{6}$
2. Substitute into Equation 2: Replace $L$ in Equation 2 with the expression derived above: $9S = 7 \left( \frac{7S + 7}{6} \right) + 21$
3. Solve for $S$: To eliminate the fraction, multiply the entire equation by 6: $6 \times 9S = 6 \times 7 \left( \frac{7S + 7}{6} \right) + 6 \times 21$ $54S = 7(7S + 7) + 126$ $54S = 49S + 49 + 126$ $54S = 49S + 175$ Subtract $49S$ from both sides: $54S - 49S = 175$ $5S = 175$ Divide by 5: $S = \frac{175}{5}$ $S = 35$
4. Solve for $L$: Substitute the value of $S = 35$ back into the expression for $L$: $L = \frac{7(35) + 7}{6}$ $L = \frac{245 + 7}{6}$ $L = \frac{252}{6}$ $L = 42$

Verification of Results

Let's check if the numbers $L=42$ and $S=35$ satisfy the original conditions.

• Condition 1 Check: $6L = 6 \times 42 = 252$. $7S + 7 = 7 \times 35 + 7 = 245 + 7 = 252$. The condition holds true.
• Condition 2 Check: $9S = 9 \times 35 = 315$. $7L + 21 = 7 \times 42 + 21 = 294 + 21 = 315$. The condition holds true.

Conclusion

Both conditions are satisfied with the numbers 42 and 35. Therefore, the two numbers are 42 and 35.