Answer :
Let's solve the problem step by step.
### Step (a): Total amount of paint used on the house
1. Convert the mixed numbers to improper fractions:
- Exterior paint: [tex]\( 4 \frac{3}{5} \)[/tex]
[tex]\[ 4 \frac{3}{5} = 4 + \frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{20 + 3}{5} = \frac{23}{5} \][/tex]
- Interior paint: [tex]\( 9 \frac{5}{6} \)[/tex]
[tex]\[ 9 \frac{5}{6} = 9 + \frac{5}{6} = \frac{9 \times 6 + 5}{6} = \frac{54 + 5}{6} = \frac{59}{6} \][/tex]
2. Find a common denominator to add the fractions:
- The least common multiple (LCM) of 5 and 6 is 30.
3. Convert fractions to have the same denominator:
- For exterior paint:
[tex]\[ \frac{23}{5} = \frac{23 \times 6}{5 \times 6} = \frac{138}{30} \][/tex]
- For interior paint:
[tex]\[ \frac{59}{6} = \frac{59 \times 5}{6 \times 5} = \frac{295}{30} \][/tex]
4. Add the fractions:
[tex]\[ \text{Total paint on house} = \frac{138}{30} + \frac{295}{30} = \frac{138 + 295}{30} = \frac{433}{30} \][/tex]
5. Convert the improper fraction back to a mixed number:
- [tex]\( \frac{433}{30} \approx 14 \frac{13}{30} \)[/tex]
So the total amount of paint used on the house is [tex]\( 14 \frac{13}{30} \)[/tex] gallons.
[tex]\[ 14 \frac{13}{30} \text{ gallons} \][/tex]
### Step (b): Total amount of paint used on the house and garage
1. Convert the additional paint used on the garage to an improper fraction:
[tex]\[ 8 \frac{4}{5} = 8 + \frac{4}{5} = \frac{8 \times 5 + 4}{5} = \frac{40 + 4}{5} = \frac{44}{5} \][/tex]
2. Convert fractions to have the same denominator:
- For house and garage total in terms of 30:
[tex]\[ \frac{44}{5} = \frac{44 \times 6}{5 \times 6} = \frac{264}{30} \][/tex]
3. Add this to the previous total:
[tex]\[ \text{Total paint on house and garage} = \frac{433}{30} + \frac{264}{30} = \frac{433 + 264}{30} = \frac{697}{30} \][/tex]
4. Convert the improper fraction back to a mixed number:
- [tex]\( \frac{697}{30} \approx 23 \frac{7}{30} \)[/tex]
So the total amount of paint used on the house and garage is [tex]\( 23 \frac{7}{30} \)[/tex] gallons.
[tex]\[ 23 \frac{7}{30} \text{ gallons} \][/tex]
### Step (c): Total cost rounding up to the next whole gallon
1. Round up the total paint amount to the next whole number:
- [tex]\( 23 \frac{7}{30} \approx 24 \)[/tex] gallons (since we need to round up)
2. Calculate the total cost:
[tex]\[ \text{Cost per gallon = } \$22 \][/tex]
[tex]\[ \text{Total cost} = 24 \text{ gallons} \times \$22 \text{ per gallon} = \$528 \][/tex]
So the total cost of the paint is [tex]\( \$528 \)[/tex].
[tex]\[ \$528 \][/tex]
### Step (a): Total amount of paint used on the house
1. Convert the mixed numbers to improper fractions:
- Exterior paint: [tex]\( 4 \frac{3}{5} \)[/tex]
[tex]\[ 4 \frac{3}{5} = 4 + \frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{20 + 3}{5} = \frac{23}{5} \][/tex]
- Interior paint: [tex]\( 9 \frac{5}{6} \)[/tex]
[tex]\[ 9 \frac{5}{6} = 9 + \frac{5}{6} = \frac{9 \times 6 + 5}{6} = \frac{54 + 5}{6} = \frac{59}{6} \][/tex]
2. Find a common denominator to add the fractions:
- The least common multiple (LCM) of 5 and 6 is 30.
3. Convert fractions to have the same denominator:
- For exterior paint:
[tex]\[ \frac{23}{5} = \frac{23 \times 6}{5 \times 6} = \frac{138}{30} \][/tex]
- For interior paint:
[tex]\[ \frac{59}{6} = \frac{59 \times 5}{6 \times 5} = \frac{295}{30} \][/tex]
4. Add the fractions:
[tex]\[ \text{Total paint on house} = \frac{138}{30} + \frac{295}{30} = \frac{138 + 295}{30} = \frac{433}{30} \][/tex]
5. Convert the improper fraction back to a mixed number:
- [tex]\( \frac{433}{30} \approx 14 \frac{13}{30} \)[/tex]
So the total amount of paint used on the house is [tex]\( 14 \frac{13}{30} \)[/tex] gallons.
[tex]\[ 14 \frac{13}{30} \text{ gallons} \][/tex]
### Step (b): Total amount of paint used on the house and garage
1. Convert the additional paint used on the garage to an improper fraction:
[tex]\[ 8 \frac{4}{5} = 8 + \frac{4}{5} = \frac{8 \times 5 + 4}{5} = \frac{40 + 4}{5} = \frac{44}{5} \][/tex]
2. Convert fractions to have the same denominator:
- For house and garage total in terms of 30:
[tex]\[ \frac{44}{5} = \frac{44 \times 6}{5 \times 6} = \frac{264}{30} \][/tex]
3. Add this to the previous total:
[tex]\[ \text{Total paint on house and garage} = \frac{433}{30} + \frac{264}{30} = \frac{433 + 264}{30} = \frac{697}{30} \][/tex]
4. Convert the improper fraction back to a mixed number:
- [tex]\( \frac{697}{30} \approx 23 \frac{7}{30} \)[/tex]
So the total amount of paint used on the house and garage is [tex]\( 23 \frac{7}{30} \)[/tex] gallons.
[tex]\[ 23 \frac{7}{30} \text{ gallons} \][/tex]
### Step (c): Total cost rounding up to the next whole gallon
1. Round up the total paint amount to the next whole number:
- [tex]\( 23 \frac{7}{30} \approx 24 \)[/tex] gallons (since we need to round up)
2. Calculate the total cost:
[tex]\[ \text{Cost per gallon = } \$22 \][/tex]
[tex]\[ \text{Total cost} = 24 \text{ gallons} \times \$22 \text{ per gallon} = \$528 \][/tex]
So the total cost of the paint is [tex]\( \$528 \)[/tex].
[tex]\[ \$528 \][/tex]
