The difference between two numbers is 16. Five times the smaller is the same as 8 less than twice the larger. Find the numbers, given that $x - y = 16$.

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05-11-2026
Mathematics

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The difference between two numbers is 16. Five times the smaller is the same as 8 less than twice the larger. Find the numbers, given that $x - y = 16$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T23:48:04-05:00

Identify the variables as $x$ (larger) and $y$ (smaller) and translate the problem into the system of equations: $x - y = 16$ and $5y = 2x - 8$.
Use substitution to solve for $y$: substitute $x = y + 16$ into the second equation, which yields $5y = 2(y+16) - 8$, resulting in $y=8$.
Determine $x$ by substituting $y=8$ into $x = y + 16$, giving $x=24$.
The final numbers are $x=24$ and $y=8$, written as $\boxed{24, 8}$.

Explanation

Problem Analysis

We are given two numbers, which we will call $x$ and $y$. Let $x$ represent the larger number and $y$ represent the smaller number. We are provided with two conditions regarding these numbers:

The difference between the numbers is 16: $x - y = 16$.
Five times the smaller number is equal to 8 less than twice the larger number: $5y = 2x - 8$.

Our objective is to find the specific values of $x$ and $y$ that satisfy both these conditions simultaneously.

Solving the System of Equations

We have a system of two linear equations:
1) $x - y = 16$
2) $5y = 2x - 8$

To solve this system, we can use the substitution method. First, we rearrange the first equation to express $x$ in terms of $y$:
\[x = y + 16\]
Now, we substitute this expression for $x$ into the second equation:
\[5y = 2(y + 16) - 8\]
Next, we expand the right side of the equation:
\[5y = 2y + 32 - 8\]
\[5y = 2y + 24\]
Now, we collect all terms involving $y$ on one side by subtracting $2y$ from both sides:
\[5y - 2y = 24\]
\[3y = 24\]
Finally, dividing both sides by 3 gives:
\[y = 8\]
With the value of $y$ determined, we substitute it back into the expression for $x$:
\[x = 8 + 16\]
\[x = 24\]
Thus, the two numbers are 24 and 8.

Conclusion

We have found that the larger number is $x = 24$ and the smaller number is $y = 8$.
Checking the conditions:

The difference between the numbers is $24 - 8 = 16$, which matches the first condition.
Five times the smaller number is $5 \times 8 = 40$, and 8 less than twice the larger number is $2(24) - 8 = 48 - 8 = 40$. Both sides are equal, matching the second condition.

The values satisfy the given conditions, and therefore the numbers are 24 and 8. The larger number is 24 and the smaller number is 8.

Examples

This type of algebraic system of equations is frequently used in resource allocation or balancing problems. For example, if you are splitting a total budget of \$160 between two marketing campaigns, where one campaign is allocated \$16 more than the other, you would set up a similar system of equations to determine the exact budget for each project. Solving these equations helps in making precise financial and management decisions.