Answer :
To solve this problem, let's go through it step by step.
### Step 1: Estimate the Difference
First, we need to round each mixed number to the nearest half to estimate the difference.
1. [tex]$7 \frac{2}{3}$[/tex]:
- [tex]$\frac{2}{3}$[/tex] is approximately [tex]$0.67$[/tex].
- If we think of [tex]$0$[/tex] as [tex]$0$[/tex], [tex]$0.25$[/tex] as [tex]$0.5/2$[/tex], [tex]$0.5$[/tex] as [tex]$1/2$[/tex], [tex]$0.75$[/tex] as [tex]$1$[/tex], and [tex]$1$[/tex] as [tex]$2$[/tex], [tex]$0.67$[/tex] is closer to [tex]$0.5$[/tex] than to [tex]$1$[/tex].
- So, [tex]$7 \frac{2}{3}$[/tex] rounds to [tex]$7.5$[/tex] (which is [tex]$7 \frac{1}{2}$[/tex] when expressed as a mixed number).
2. [tex]$2 \frac{14}{15}$[/tex]:
- [tex]$\frac{14}{15}$[/tex] is approximately [tex]$0.93$[/tex], which is closer to [tex]$1$[/tex] than to [tex]$0.5$[/tex].
- So, [tex]$2 \frac{14}{15}$[/tex] rounds to [tex]$3$[/tex].
Thus, our estimated difference is:
[tex]\[ 7.5 - 3 = 4.5 \][/tex]
### Step 2: Find the Actual Difference
Let's convert the mixed numbers to improper fractions and then find the difference:
1. Convert [tex]$7 \frac{2}{3}$[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[ 7 \times 3 + 2 = 21 + 2 = 23 \][/tex]
- So, the improper fraction is [tex]$\frac{23}{3}$[/tex].
2. Convert [tex]$2 \frac{14}{15}$[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[ 2 \times 15 + 14 = 30 + 14 = 44 \][/tex]
- So, the improper fraction is [tex]$\frac{44}{15}$[/tex].
3. Find the difference of these fractions:
- To subtract these, we need a common denominator. The least common multiple (LCM) of 3 and 15 is 15.
- Convert [tex]$\frac{23}{3}$[/tex] to [tex]$\frac{115}{15}$[/tex] (since [tex]$23 \times 5 = 115$[/tex]).
- The difference between [tex]$\frac{115}{15}$[/tex] and [tex]$\frac{44}{15}$[/tex] is:
[tex]\[ \frac{115}{15} - \frac{44}{15} = \frac{71}{15} \][/tex]
This fraction is already in its simplest form.
### Final Answer
The estimated difference is [tex]$4.5$[/tex], and the actual difference in simplest form is:
[tex]\[ 7 \frac{2}{3} - 2 \frac{14}{15} = \frac{71}{15} \][/tex]
### Step 1: Estimate the Difference
First, we need to round each mixed number to the nearest half to estimate the difference.
1. [tex]$7 \frac{2}{3}$[/tex]:
- [tex]$\frac{2}{3}$[/tex] is approximately [tex]$0.67$[/tex].
- If we think of [tex]$0$[/tex] as [tex]$0$[/tex], [tex]$0.25$[/tex] as [tex]$0.5/2$[/tex], [tex]$0.5$[/tex] as [tex]$1/2$[/tex], [tex]$0.75$[/tex] as [tex]$1$[/tex], and [tex]$1$[/tex] as [tex]$2$[/tex], [tex]$0.67$[/tex] is closer to [tex]$0.5$[/tex] than to [tex]$1$[/tex].
- So, [tex]$7 \frac{2}{3}$[/tex] rounds to [tex]$7.5$[/tex] (which is [tex]$7 \frac{1}{2}$[/tex] when expressed as a mixed number).
2. [tex]$2 \frac{14}{15}$[/tex]:
- [tex]$\frac{14}{15}$[/tex] is approximately [tex]$0.93$[/tex], which is closer to [tex]$1$[/tex] than to [tex]$0.5$[/tex].
- So, [tex]$2 \frac{14}{15}$[/tex] rounds to [tex]$3$[/tex].
Thus, our estimated difference is:
[tex]\[ 7.5 - 3 = 4.5 \][/tex]
### Step 2: Find the Actual Difference
Let's convert the mixed numbers to improper fractions and then find the difference:
1. Convert [tex]$7 \frac{2}{3}$[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[ 7 \times 3 + 2 = 21 + 2 = 23 \][/tex]
- So, the improper fraction is [tex]$\frac{23}{3}$[/tex].
2. Convert [tex]$2 \frac{14}{15}$[/tex] to an improper fraction:
- Multiply the whole number by the denominator and add the numerator:
[tex]\[ 2 \times 15 + 14 = 30 + 14 = 44 \][/tex]
- So, the improper fraction is [tex]$\frac{44}{15}$[/tex].
3. Find the difference of these fractions:
- To subtract these, we need a common denominator. The least common multiple (LCM) of 3 and 15 is 15.
- Convert [tex]$\frac{23}{3}$[/tex] to [tex]$\frac{115}{15}$[/tex] (since [tex]$23 \times 5 = 115$[/tex]).
- The difference between [tex]$\frac{115}{15}$[/tex] and [tex]$\frac{44}{15}$[/tex] is:
[tex]\[ \frac{115}{15} - \frac{44}{15} = \frac{71}{15} \][/tex]
This fraction is already in its simplest form.
### Final Answer
The estimated difference is [tex]$4.5$[/tex], and the actual difference in simplest form is:
[tex]\[ 7 \frac{2}{3} - 2 \frac{14}{15} = \frac{71}{15} \][/tex]
