Solved on Mar 19, 2024

Solve the mixed fraction addition problem: $37 \frac{3}{4} + \frac{8}{9} + 5 \frac{1}{12}$ .

STEP 1

Assumptions
1. We are adding three mixed numbers: $37 \frac{3}{4}$ , $\frac{8}{9}$ , and $5 \frac{1}{12}$ .
2. We need to find a common denominator to add the fractions.
3. The sum of the whole numbers will be added separately from the sum of the fractions.

STEP 2

First, let's convert the mixed numbers to improper fractions to make it easier to find a common denominator and add them.
For $37 \frac{3}{4}$ , we convert it to an improper fraction by multiplying the whole number by the denominator of the fraction and then adding the numerator.
$37 \frac{3}{4} = \frac{37 \times 4 + 3}{4}$

STEP 3

Calculate the improper fraction for $37 \frac{3}{4}$ .
$37 \frac{3}{4} = \frac{148 + 3}{4}$ $37 \frac{3}{4} = \frac{151}{4}$

STEP 4

For $5 \frac{1}{12}$ , we convert it to an improper fraction using the same method.
$5 \frac{1}{12} = \frac{5 \times 12 + 1}{12}$

STEP 5

Calculate the improper fraction for $5 \frac{1}{12}$ .
$5 \frac{1}{12} = \frac{60 + 1}{12}$ $5 \frac{1}{12} = \frac{61}{12}$

STEP 6

Now we have three fractions: $\frac{151}{4}$ , $\frac{8}{9}$ , and $\frac{61}{12}$ . We need to find a common denominator to add these fractions.
The common denominator for 4, 9, and 12 is their least common multiple (LCM).

STEP 7

Find the LCM of 4, 9, and 12.
The prime factorization of the denominators: - $4 = 2^2$ - $9 = 3^2$ - $12 = 2^2 \times 3$
The LCM is the product of the highest powers of all prime factors: $LCM = 2^2 \times 3^2$

STEP 8

Calculate the LCM.
$LCM = 4 \times 9$ $LCM = 36$

STEP 9

Now that we have the common denominator of 36, we convert each fraction to have this common denominator.
For $\frac{151}{4}$ , we find the equivalent fraction with a denominator of 36.
$\frac{151}{4} \times \frac{9}{9} = \frac{151 \times 9}{36}$

STEP 10

Calculate the new numerator for $\frac{151}{4}$ .
$151 \times 9 = 1359$

STEP 11

Write the equivalent fraction for $\frac{151}{4}$ with the common denominator.
$\frac{151}{4} = \frac{1359}{36}$

STEP 12

For $\frac{8}{9}$ , we find the equivalent fraction with a denominator of 36.
$\frac{8}{9} \times \frac{4}{4} = \frac{8 \times 4}{36}$

STEP 13

Calculate the new numerator for $\frac{8}{9}$ .
$8 \times 4 = 32$

STEP 14

Write the equivalent fraction for $\frac{8}{9}$ with the common denominator.
$\frac{8}{9} = \frac{32}{36}$

STEP 15

For $\frac{61}{12}$ , we find the equivalent fraction with a denominator of 36.
$\frac{61}{12} \times \frac{3}{3} = \frac{61 \times 3}{36}$

STEP 16

Calculate the new numerator for $\frac{61}{12}$ .
$61 \times 3 = 183$

STEP 17

Write the equivalent fraction for $\frac{61}{12}$ with the common denominator.
$\frac{61}{12} = \frac{183}{36}$

STEP 18

Now we add the fractions with the common denominator.
$\frac{1359}{36} + \frac{32}{36} + \frac{183}{36}$

STEP 19

Add the numerators.
$1359 + 32 + 183 = 1574$

STEP 20

Write the sum of the fractions.
$\frac{1359}{36} + \frac{32}{36} + \frac{183}{36} = \frac{1574}{36}$

STEP 21

Simplify the fraction $\frac{1574}{36}$ if possible.
$\frac{1574}{36} = \frac{437}{9}$

STEP 22

Convert the improper fraction $\frac{437}{9}$ back to a mixed number.
$\frac{437}{9} = 48 \frac{5}{9}$

STEP 23

Now we combine the whole number part of the mixed number with any whole numbers from the original problem. In this case, there are no additional whole numbers to add.

STEP 24

Write the final answer as a mixed number.
$37 \frac{3}{4} + \frac{8}{9} + 5 \frac{1}{12} = 48 \frac{5}{9}$
The sum of the given numbers is $48 \frac{5}{9}$ .

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Solve the mixed fraction addition problem: 3734+89+511237 \frac{3}{4} + \frac{8}{9} + 5 \frac{1}{12}3743​+98​+5121​.

STEP 1

Assumptions1. We are adding three mixed numbers: 373437 \frac{3}{4}3743​, 89\frac{8}{9}98​, and 51125 \frac{1}{12}5121​.2. We need to find a common denominator to add the fractions.3. The sum of the whole numbers will be added separately from the sum of the fractions.

STEP 2

STEP 3

Calculate the improper fraction for 373437 \frac{3}{4}3743​.3734=148+3437 \frac{3}{4} = \frac{148 + 3}{4}3743​=4148+3​ 3734=151437 \frac{3}{4} = \frac{151}{4}3743​=4151​

STEP 4

For 51125 \frac{1}{12}5121​, we convert it to an improper fraction using the same method.5112=5×12+1125 \frac{1}{12} = \frac{5 \times 12 + 1}{12}5121​=125×12+1​

STEP 5

Calculate the improper fraction for 51125 \frac{1}{12}5121​.5112=60+1125 \frac{1}{12} = \frac{60 + 1}{12}5121​=1260+1​ 5112=61125 \frac{1}{12} = \frac{61}{12}5121​=1261​

STEP 6

Now we have three fractions: 1514\frac{151}{4}4151​, 89\frac{8}{9}98​, and 6112\frac{61}{12}1261​. We need to find a common denominator to add these fractions.The common denominator for 4, 9, and 12 is their least common multiple (LCM).

STEP 7

Find the LCM of 4, 9, and 12.The prime factorization of the denominators: - 4=224 = 2^24=22 - 9=329 = 3^29=32 - 12=22×312 = 2^2 \times 312=22×3The LCM is the product of the highest powers of all prime factors: LCM=22×32LCM = 2^2 \times 3^2LCM=22×32

STEP 8

Calculate the LCM.LCM=4×9LCM = 4 \times 9LCM=4×9 LCM=36LCM = 36LCM=36

STEP 9

Now that we have the common denominator of 36, we convert each fraction to have this common denominator.For 1514\frac{151}{4}4151​, we find the equivalent fraction with a denominator of 36.1514×99=151×936\frac{151}{4} \times \frac{9}{9} = \frac{151 \times 9}{36}4151​×99​=36151×9​

STEP 10

Calculate the new numerator for 1514\frac{151}{4}4151​.151×9=1359151 \times 9 = 1359151×9=1359

STEP 11

Write the equivalent fraction for 1514\frac{151}{4}4151​ with the common denominator.1514=135936\frac{151}{4} = \frac{1359}{36}4151​=361359​

STEP 12

For 89\frac{8}{9}98​, we find the equivalent fraction with a denominator of 36.89×44=8×436\frac{8}{9} \times \frac{4}{4} = \frac{8 \times 4}{36}98​×44​=368×4​

STEP 13

Calculate the new numerator for 89\frac{8}{9}98​.8×4=328 \times 4 = 328×4=32

STEP 14

Write the equivalent fraction for 89\frac{8}{9}98​ with the common denominator.89=3236\frac{8}{9} = \frac{32}{36}98​=3632​

STEP 15

For 6112\frac{61}{12}1261​, we find the equivalent fraction with a denominator of 36.6112×33=61×336\frac{61}{12} \times \frac{3}{3} = \frac{61 \times 3}{36}1261​×33​=3661×3​

STEP 16

Calculate the new numerator for 6112\frac{61}{12}1261​.61×3=18361 \times 3 = 18361×3=183

STEP 17

Write the equivalent fraction for 6112\frac{61}{12}1261​ with the common denominator.6112=18336\frac{61}{12} = \frac{183}{36}1261​=36183​

STEP 18

Now we add the fractions with the common denominator.135936+3236+18336\frac{1359}{36} + \frac{32}{36} + \frac{183}{36}361359​+3632​+36183​

STEP 19

Add the numerators.1359+32+183=15741359 + 32 + 183 = 15741359+32+183=1574

STEP 20

Write the sum of the fractions.135936+3236+18336=157436\frac{1359}{36} + \frac{32}{36} + \frac{183}{36} = \frac{1574}{36}361359​+3632​+36183​=361574​

STEP 21

Simplify the fraction 157436\frac{1574}{36}361574​ if possible.157436=4379\frac{1574}{36} = \frac{437}{9}361574​=9437​

STEP 22

Convert the improper fraction 4379\frac{437}{9}9437​ back to a mixed number.4379=4859\frac{437}{9} = 48 \frac{5}{9}9437​=4895​

STEP 23

Now we combine the whole number part of the mixed number with any whole numbers from the original problem. In this case, there are no additional whole numbers to add.

STEP 24