Question

Noa was walking in the forest and measured the circumference of two trees that he found. The first tree measured 37 and StartFraction 5 over 8 EndFraction inches around, and the second tree measured 45 and one-third inches around. Noa wanted to find the difference of the circumferences of the two trees. He recorded his steps in the table.

Steps
Noa’s work

Step 1

45 and one-third minus 37 and StartFraction 5 over 8 EndFraction

Step 2

45 and StartFraction 8 over 24 EndFraction minus 37 and StartFraction 15 over 24 EndFraction

Step 3

44 and StartFraction 32 over 24 EndFraction minus 37 and StartFraction 15 over 24 EndFraction

Step 4

44 and StartFraction 32 over 24 EndFraction minus 37 and StartFraction 15 over 24 EndFraction = 8 and StartFraction 17 over 24 EndFraction

In which step did Noa first make an error?
Step 1
Step 2
Step 3
Step 4

Answer 1

To determine where Noa first made an error, let's analyze each step of his work:

**Step 1:**
Noa computes:
\[ 45 \frac{1}{3} - 37 \frac{5}{8} \]

This is correct as he properly states the subtraction of the two numbers.

**Step 2:**
He converts the mixed numbers to improper fractions:
- \( 45 \frac{1}{3} = \frac{136}{3} \)
- \( 37 \frac{5}{8} = \frac{301}{8} \)

Then he attempts to find a common denominator, which is 24. However, he seems to incorrectly convert the fractions:
- \( 45 \frac{1}{3} \) should be converted as follows:
\[
45 \frac{1}{3} = 45 + \frac{1}{3} = \frac{135}{3} + \frac{1}{3} = \frac{136}{3}
\]
which correctly converts to:
\[
\frac{136}{3} \times \frac{8}{8} = \frac{1088}{24}
\]
- \( 37 \frac{5}{8} \):
\[
37 \frac{5}{8} = 37 + \frac{5}{8} = \frac{296}{8} + \frac{5}{8} = \frac{301}{8}
\]
which converts to:
\[
\frac{301}{8} \times \frac{3}{3} = \frac{903}{24}
\]

Therefore, he mistakenly represents:
\[
45 \frac{1}{3} = 45 \frac{8}{24}
\]
and:
\[
37 \frac{5}{8} = 37 \frac{15}{24}
\]

These conversions are incorrect in their representation as fractions, where he converts the whole numbers improperly.

**Step 3:**
In this step he states:
\[
44 \frac{32}{24} - 37 \frac{15}{24}
\]
This itself doesn't make sense as he doesn’t keep the proper fractions and their conversions correct from Step 2.

**Step 4:**
The subtraction is done here, but since the previous steps were flawed, the entire conclusion is based on incorrect fractions.

**Final decision:**
Noa first made an error in **Step 2** when trying to convert the fractions into a common denominator format. Thus the answer is:
**Step 2**.