Question

Kristina, Liz, and Heather all made cookies. Kristina made 2.4 dozen less than 6 times as many dozen as Heather. Liz made 1.8 dozen more than 5 times as many dozen as Heather. If Kristina and Liz made the same number of cookies, how many dozen cookies did Heather make?

Answer 1

Let's assign a variable to represent the number of dozens of cookies Heather made. We'll call this variable \( H \).

According to the problem:
- Kristina made \( 6 \times H - 2.4 \) dozen cookies.
- Liz made \( 5 \times H + 1.8 \) dozen cookies.

We are also given that Kristina and Liz made the same number of cookies, so we can set these two expressions equal to each other to find the value of \( H \):

\[ 6H - 2.4 = 5H + 1.8 \]

Now, we'll solve this equation for \( H \).

Step 1: Isolate the variable \( H \) on one side.
To do this, we'll first subtract \( 5H \) from both sides of the equation:

\[ 6H - 5H - 2.4 = 5H - 5H + 1.8 \]
\[ H - 2.4 = 1.8 \]

Step 2: Now we need to get \( H \) alone, so we'll add \( 2.4 \) to both sides:

\[ H - 2.4 + 2.4 = 1.8 + 2.4 \]
\[ H = 4.2 \]

So, Heather made \( 4.2 \) dozen cookies.

To ensure this is the correct number of dozens Heather made, we can plug this value back into the original expressions for Kristina and Liz's cookies to verify that they made the same amount:

Kristina: \( 6 \times 4.2 - 2.4 = 25.2 - 2.4 = 22.8 \) dozen cookies
Liz: \( 5 \times 4.2 + 1.8 = 21 + 1.8 = 22.8 \) dozen cookies

Both Kristina and Liz made 22.8 dozen cookies, confirming our solution that Heather made 4.2 dozen cookies.

Answer 2

Based on the number of cookies each girl made and if Kristina and Liz made the same number of cookies, then Heather made 4.2 dozen cookies.

Explanation:

In order to find the number of cookies that Heather made, you need to set up two equations that represent the number of cookies that Kristina and Liz each made. Since Kristina made 2.4 dozen less than 6 times as many dozen as Heather, we can represent that with the equation 6c - 2.4. Since Liz made 1.8 dozen more than 5 times as many dozen as Heather, we can represent that with the equation 5c + 1.8. We can set both equations equal to each other since Liz and Kristina made the same number, so 6c - 2.4 = 5c + 1.8. Solving for 'c', which represents how many dozen cookies Heather made, we get 4.2