Question
A full bucket of paint holds 2 1/2 gallons of paint. Jan has 3 3/8 buckets of paint. She followed these steps to calculate how many gallons of paint she has, but she made a mistake in her work.
First, write an equation to represent the problem. Use p to represent how many gallons of paint there are.
2 1/2 x 3 3/8 = p
Next, use the distributive property to multiply. Write each mixed number as an addition expression. Multiply the whole number, and multiply the fractions.
2 1/2 x 3 3/8 = (2 + 1/2) x (3 + 3/8)
= (2x3) + (1/2 x 3/8)
= 6 + 3/16
Last, add the two products
p= 6 + 3/16 = 6 3/16
So, there are 6 3/16 gallons of paint
Select the statement that best describes Jan's mistake. Then select the total amount of paint she has
a. She forgot to multiply 2 by 3/8 and 3 by 1/2
b. She should have added 2 1/2 and 3 3/8 instead of multiplying them
c. She forgot to change 1/2 and 3/8 to have like denominators
d. 6 4/8
e. 7 14/16
f. 5 7/8
g. 8 7/16
First, write an equation to represent the problem. Use p to represent how many gallons of paint there are.
2 1/2 x 3 3/8 = p
Next, use the distributive property to multiply. Write each mixed number as an addition expression. Multiply the whole number, and multiply the fractions.
2 1/2 x 3 3/8 = (2 + 1/2) x (3 + 3/8)
= (2x3) + (1/2 x 3/8)
= 6 + 3/16
Last, add the two products
p= 6 + 3/16 = 6 3/16
So, there are 6 3/16 gallons of paint
Select the statement that best describes Jan's mistake. Then select the total amount of paint she has
a. She forgot to multiply 2 by 3/8 and 3 by 1/2
b. She should have added 2 1/2 and 3 3/8 instead of multiplying them
c. She forgot to change 1/2 and 3/8 to have like denominators
d. 6 4/8
e. 7 14/16
f. 5 7/8
g. 8 7/16
Answers
GPT-4o mini
Let's first analyze Jan's calculations step by step to identify the mistake.
The original problem is to calculate the total gallons of paint using the equation:
\[
p = 2\frac{1}{2} \times 3\frac{3}{8}
\]
Jan expresses the mixed numbers as follows:
\[
2\frac{1}{2} = 2 + \frac{1}{2} \quad \text{and} \quad 3\frac{3}{8} = 3 + \frac{3}{8}
\]
Then, she uses the distributive property:
\[
(2 + \frac{1}{2}) \times (3 + \frac{3}{8})
\]
Expanding this gives:
\[
2 \times 3 + 2 \times \frac{3}{8} + \frac{1}{2} \times 3 + \frac{1}{2} \times \frac{3}{8}
\]
Calculating this correctly:
1. \(2 \times 3 = 6\)
2. \(2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4}\)
3. \(\frac{1}{2} \times 3 = \frac{3}{2}\)
4. \(\frac{1}{2} \times \frac{3}{8} = \frac{3}{16}\)
Now, we need to add these products:
\[
6 + \frac{3}{4} + \frac{3}{2} + \frac{3}{16}
\]
First, let's convert everything to have a common denominator of 16:
\[
6 = \frac{96}{16}, \quad \frac{3}{4} = \frac{12}{16}, \quad \frac{3}{2} = \frac{24}{16}, \quad \frac{3}{16} = \frac{3}{16}
\]
Adding these together:
\[
\frac{96}{16} + \frac{12}{16} + \frac{24}{16} + \frac{3}{16} = \frac{135}{16}
\]
Now convert this improper fraction to a mixed number:
\[
\frac{135}{16} = 8\frac{7}{16}
\]
Therefore, Jan should have calculated that she has:
\[
p = 8\frac{7}{16}
\]
Now, let's identify Jan's mistake from the options provided:
- **a.** She forgot to multiply 2 by 3/8 and 3 by 1/2 (this is true; she did not account for all products when distributing).
- **b.** She should have added \(2\frac{1}{2}\) and \(3\frac{3}{8}\) instead of multiplying them (this is not true; multiplication is the correct operation here).
- **c.** She forgot to change 1/2 and 3/8 to have like denominators (not applicable; she didn't use the fractions correctly in distribution).
- **d.** \(6 \frac{4}{8} \) (this isn't correct).
- **e.** \(7 \frac{14}{16}\) (this isn't correct).
- **f.** \(5 \frac{7}{8}\) (this isn't correct).
- **g.** \(8 \frac{7}{16}\) (this is the correct total amount of paint).
The best statement that describes her mistake is **a**.
The total amount of paint Jan has is **g. 8 7/16**.
The original problem is to calculate the total gallons of paint using the equation:
\[
p = 2\frac{1}{2} \times 3\frac{3}{8}
\]
Jan expresses the mixed numbers as follows:
\[
2\frac{1}{2} = 2 + \frac{1}{2} \quad \text{and} \quad 3\frac{3}{8} = 3 + \frac{3}{8}
\]
Then, she uses the distributive property:
\[
(2 + \frac{1}{2}) \times (3 + \frac{3}{8})
\]
Expanding this gives:
\[
2 \times 3 + 2 \times \frac{3}{8} + \frac{1}{2} \times 3 + \frac{1}{2} \times \frac{3}{8}
\]
Calculating this correctly:
1. \(2 \times 3 = 6\)
2. \(2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4}\)
3. \(\frac{1}{2} \times 3 = \frac{3}{2}\)
4. \(\frac{1}{2} \times \frac{3}{8} = \frac{3}{16}\)
Now, we need to add these products:
\[
6 + \frac{3}{4} + \frac{3}{2} + \frac{3}{16}
\]
First, let's convert everything to have a common denominator of 16:
\[
6 = \frac{96}{16}, \quad \frac{3}{4} = \frac{12}{16}, \quad \frac{3}{2} = \frac{24}{16}, \quad \frac{3}{16} = \frac{3}{16}
\]
Adding these together:
\[
\frac{96}{16} + \frac{12}{16} + \frac{24}{16} + \frac{3}{16} = \frac{135}{16}
\]
Now convert this improper fraction to a mixed number:
\[
\frac{135}{16} = 8\frac{7}{16}
\]
Therefore, Jan should have calculated that she has:
\[
p = 8\frac{7}{16}
\]
Now, let's identify Jan's mistake from the options provided:
- **a.** She forgot to multiply 2 by 3/8 and 3 by 1/2 (this is true; she did not account for all products when distributing).
- **b.** She should have added \(2\frac{1}{2}\) and \(3\frac{3}{8}\) instead of multiplying them (this is not true; multiplication is the correct operation here).
- **c.** She forgot to change 1/2 and 3/8 to have like denominators (not applicable; she didn't use the fractions correctly in distribution).
- **d.** \(6 \frac{4}{8} \) (this isn't correct).
- **e.** \(7 \frac{14}{16}\) (this isn't correct).
- **f.** \(5 \frac{7}{8}\) (this isn't correct).
- **g.** \(8 \frac{7}{16}\) (this is the correct total amount of paint).
The best statement that describes her mistake is **a**.
The total amount of paint Jan has is **g. 8 7/16**.