Math Problem Statement

There are two numbers. The first number minus the second number is 15. One-third of the sum of the numbers is one-quarter of the first number. What are the two numbers?

Solution

The problem from the image states:

b. There are two numbers. The first number minus the second number is 15. One-third of the sum of the numbers is one-quarter of the first number. What are the two numbers?

Let's define:

  • The first number as xx
  • The second number as yy

We are given two conditions:

  1. The first number minus the second number is 15:
    xy=15(Equation 1)x - y = 15 \quad \text{(Equation 1)}
  2. One-third of the sum of the numbers is one-quarter of the first number:
    13(x+y)=14x(Equation 2)\frac{1}{3}(x + y) = \frac{1}{4}x \quad \text{(Equation 2)}

Solving the system:

From Equation 1, we can solve for yy: y=x15y = x - 15

Substitute this into Equation 2: 13(x+(x15))=14x\frac{1}{3}(x + (x - 15)) = \frac{1}{4}x Simplify the left side: 13(2x15)=14x\frac{1}{3}(2x - 15) = \frac{1}{4}x Multiply both sides by 12 to eliminate the denominators: 12×13(2x15)=12×14x12 \times \frac{1}{3}(2x - 15) = 12 \times \frac{1}{4}x 4(2x15)=3x4(2x - 15) = 3x Expand and solve: 8x60=3x8x - 60 = 3x 8x3x=608x - 3x = 60 5x=605x = 60 x=12x = 12

Now, substitute x=12x = 12 back into Equation 1: 12y=1512 - y = 15 y=1215=3y = 12 - 15 = -3

Thus, the two numbers are 12 and -3.

Would you like more details or have any questions?

Related Questions:

  1. How would the solution change if the sum condition were different?
  2. Can you graphically represent the two equations to find the intersection?
  3. What would happen if the difference between the numbers was another value, like 20?
  4. Can you create another riddle using similar relationships between two numbers?
  5. How would you set up the problem if the sum of the numbers was twice the first number?

Tip: Always check your solutions by substituting the values back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

x - y = 15
(1/3)(x + y) = (1/4)x

Theorems

Equation manipulation and substitution method

Suitable Grade Level

Grades 7-9