Question

Kathleen likes to knit hats and mittens for friends and family. Last fall, she knitted 5 hats and 5 pairs of mittens, which took a total of 105 hours. This fall, she knitted 5 hats and 4 pairs of mittens, which took a total of 91 hours. If each hat takes the same amount of time and each pair of mittens takes the same time, how long does it take Kathleen to knit each item?

Answer 1

- It takes Kathleen 7 hours to knit each hat.

- It takes Kathleen 14 hours to knit each pair of mittens.

Explanation:

Kathleen knitted 5 hats and 5 pairs of mittens last fall, which took a total of 105 hours. This can be expressed as:

5x + 5y = 105 (Equation 1)

Similarly, Kathleen knitted 5 hats and 4 pairs of mittens this fall, which took a total of 91 hours. This can be expressed as:

5x + 4y = 91 (Equation 2)

First, let's multiply Equation 2 by 5 to make the coefficients of x the same in both equations:

25x + 20y = 455 (Equation 3)

Next, let's subtract Equation 1 from Equation 3 to eliminate the x term:

25x + 20y - (5x + 5y) = 455 - 105

Simplifying the equation:

20x + 15y = 350 (Equation 4)

Now, we have a system of two equations:

5x + 5y = 105 (Equation 1)

20x + 15y = 350 (Equation 4)

Let's solve it using the elimination method. We'll multiply Equation 1 by 4 to make the coefficients of y the same in both equations:

20x + 20y = 420 (Equation 5)

Now, we can subtract Equation 5 from Equation 4 to eliminate the y term:

20x + 15y - (20x + 20y) = 350 - 420

Simplifying the equation:

-5y = -70

Dividing both sides of the equation by -5:

y = 14

Now, we can substitute the value of y back into Equation 1 to find the value of x:

5x + 5(14) = 105

5x + 70 = 105

5x = 35

x = 7