Question

tom drew a line that was $2\frac{5}{8}$ inches long. if he drew a second line that was $1\frac{2}{3}$ inches long, what is the difference between the length of the two lines? ( ) inches

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \( 2\frac{5}{8} \) to an improper fraction. The formula for converting a mixed number \( a\frac{b}{c} \) to an improper fraction is \( \frac{a\times c + b}{c} \). So for \( 2\frac{5}{8} \), we have \( \frac{2\times8 + 5}{8}=\frac{16 + 5}{8}=\frac{21}{8} \).

Next, convert \( 1\frac{2}{3} \) to an improper fraction. Using the same formula, \( \frac{1\times3+2}{3}=\frac{3 + 2}{3}=\frac{5}{3} \).

Step2: Find a common denominator

To subtract the two fractions, we need a common denominator. The least common denominator of 8 and 3 is \( 8\times3 = 24 \).

Convert \( \frac{21}{8} \) to a fraction with denominator 24: \( \frac{21}{8}=\frac{21\times3}{8\times3}=\frac{63}{24} \).

Convert \( \frac{5}{3} \) to a fraction with denominator 24: \( \frac{5}{3}=\frac{5\times8}{3\times8}=\frac{40}{24} \).

Step3: Subtract the fractions

Now subtract the two fractions: \( \frac{63}{24}-\frac{40}{24}=\frac{63 - 40}{24}=\frac{23}{24} \). Wait, no, wait. Wait, the first mixed number is \( 2\frac{5}{8} \) and the second is \( 1\frac{2}{3} \). Wait, maybe I made a mistake in the first number. Wait, the first line is \( 2\frac{5}{8} \) inches, the second is \( 1\frac{2}{3} \) inches. So the difference is \( 2\frac{5}{8}-1\frac{2}{3} \).

Let's redo the subtraction of mixed numbers. First, subtract the whole numbers: \( 2 - 1 = 1 \). Then subtract the fractions: \( \frac{5}{8}-\frac{2}{3} \). Find common denominator, which is 24. \( \frac{5}{8}=\frac{15}{24} \), \( \frac{2}{3}=\frac{16}{24} \). But \( \frac{15}{24}-\frac{16}{24} \) is negative, so we need to borrow 1 from the whole number part. So \( 2\frac{5}{8}=1 + 1+\frac{5}{8}=1+\frac{8}{8}+\frac{5}{8}=1\frac{13}{8} \). Wait, no, better to convert both to improper fractions. \( 2\frac{5}{8}=\frac{21}{8} \), \( 1\frac{2}{3}=\frac{5}{3} \). Now, \( \frac{21}{8}-\frac{5}{3} \). Common denominator 24: \( \frac{21\times3}{24}-\frac{5\times8}{24}=\frac{63}{24}-\frac{40}{24}=\frac{23}{24} \). Wait, that can't be, because 2 - 1 is 1, and then the fraction part: 5/8 - 2/3. Wait, 5/8 is 0.625, 2/3 is approximately 0.666, so 0.625 - 0.666 is negative, so we need to borrow 1 from the whole number. So 2 becomes 1, and we add 1 (which is 8/8) to 5/8, making 13/8. Then subtract the fractions: 13/8 - 2/3. Convert to 24 denominator: 13/8 = 39/24, 2/3 = 16/24. 39/24 - 16/24 = 23/24. Then add the whole number part: 1 + 23/24? Wait, no, the whole number part after borrowing is 1 (since we took 1 from 2, making it 1), then we have 13/8 - 2/3. Wait, I think I messed up. Let's do it again.

Mixed number subtraction: \( 2\frac{5}{8}-1\frac{2}{3} \).

Step 1: Subtract the whole numbers: 2 - 1 = 1.

Step 2: Subtract the fractions: \( \frac{5}{8}-\frac{2}{3} \).

Find a common denominator, which is 24.

\( \frac{5}{8}=\frac{5\times3}{8\times3}=\frac{15}{24} \)

\( \frac{2}{3}=\frac{2\times8}{3\times8}=\frac{16}{24} \)

Since \( \frac{15}{24}<\frac{16}{24} \), we need to borrow 1 from the whole number part. So we take 1 from the 2 (making it 1) and add it to the fraction part: \( 2\frac{5}{8}=1 + 1+\frac{5}{8}=1+\frac{8}{8}+\frac{5}{8}=1\frac{13}{8} \)

Now, subtract the mixed numbers: \( 1\frac{13}{8}-1\frac{2}{3} \)

Subtract the whole numbers: 1 - 1 = 0

Subtract the fractions: \( \frac{13}{8}-\frac{2}{3} \)

Convert to common denominator 24:

\( \frac{13}{8}=\frac{13\times3}{8\times3}=\frac{39}{24} \)

\( \frac{2}{3}=\frac{2\times8}{3\times8}=\frac{16}{24} \)

Subtract: \( \frac{39}{24}-\frac{16}{24}=\frac{23}{24} \)

So the difference is \( \frac{23}{24} \) inches? Wait, no, that seems small. Wait, 2 5/8 is 2.625, 1 2/3 is approximately 1.666, so the difference is 2.625 - 1.666 ≈ 0.958, and 23/24 ≈ 0.958, so that's correct.

Wait, maybe I made a mistake in the initial step. Let's check with decimals. 2 5/8 = 2 + 5÷8 = 2.625. 1 2/3 = 1 + 2÷3 ≈ 1.6667. Subtract: 2.625 - 1.6667 ≈ 0.9583, which is 23/24 ≈ 0.9583. So that's correct.

Answer:

\( \frac{23}{24} \)