A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?
The numbers are 7 and 35.
Explanation
To solve this, we translate the word problem into a linear equation. Let the smaller number be x, making the larger number 5x. When 21 is added to both, the new numbers are x + 21 and 5x + 21. The problem states the larger new number becomes twice the smaller new number, giving the equation 5x + 21 = 2(x + 21). Solving this linear equation gives x = 7, meaning the two numbers are 7 and 35.
Solution Steps
Step 1: Define variables. Let the smaller number be x. Since a positive number is 5 times another number, the larger number is 5x.
Step 2: Form the equation. Adding 21 to both numbers, the new numbers are (x + 21) and (5x + 21). The larger new number becomes twice the smaller new number, so: 5x + 21 = 2(x + 21).
Step 3: Solve the linear equation. Expanding the right side: 5x + 21 = 2x + 42. Transposing terms: 5x - 2x = 42 - 21, which simplifies to 3x = 21. Thus, x = 7.
Step 4: Find both numbers. The smaller number is x = 7. The larger number is 5x = = 35. The numbers are 7 and 35.