Solved on Nov 22, 2023

Evaluate the following arithmetic expressions: $\frac{1}{6}+\frac{3}{8} \times \frac{7}{9}$ , $\frac{8}{9} \times\left(\frac{1}{3}+\frac{5}{9}\right)$ , $\frac{8}{9} \div \frac{4}{9}+\frac{3}{8}$ , $-\frac{1}{2}-(-\frac{3}{1})$ , $\frac{1}{4}+\frac{3}{8} \times \frac{1}{3}$ , $\frac{5}{6} \div \frac{3}{4}-\frac{1}{5}$ , $\left(\frac{2}{9}+\frac{8}{9}\right) \times \frac{1}{6}$ , $\left(\frac{1}{2}+\frac{2}{5}\right) \times \frac{7}{8}$ , $\frac{2}{3} \div \frac{8}{9}+\frac{5}{9}$ .

STEP 1

Assumptions1. The operations must be performed in the correct order, following the BIDMAS/BODMAS rule (Brackets, Indices/Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). . The fractions are in simplest form.

STEP 2

Let's solve the first expression. According to BIDMAS, multiplication should be performed before addition.
$\frac{1}{6}+\frac{}{8} \times \frac{7}{9}$

STEP 3

Perform the multiplication operation.
$\frac{1}{6}+\frac{21}{72}$

STEP 4

Now, perform the addition operation.
$\frac{1}{6}+\frac{7}{24}$

STEP 5

To add these fractions, we need to find a common denominator. The least common multiple of and24 is24.
$\frac{4}{24}+\frac{7}{24}$

STEP 6

Now, add the fractions.
$\frac{11}{24}$

STEP 7

Now, let's solve the second expression. According to BIDMAS, operations inside brackets should be performed first.
$\frac{}{9} \times\left(\frac{1}{3}+\frac{5}{9}\right)$

STEP 8

Perform the addition operation inside the brackets.
$\frac{8}{} \times \frac{14}{27}$

STEP 9

Now, perform the multiplication operation.
$\frac{112}{243}$

STEP 10

Now, let's solve the third expression. According to BIDMAS, division should be performed before addition.
$\frac{8}{9} \div \frac{4}{9}+\frac{3}{8}$

STEP 11

Perform the division operation.
$+\frac{3}{8}$

STEP 12

Now, perform the addition operation.
$2\frac{}{8}$

STEP 13

Now, let's solve the fourth expression. According to BIDMAS, subtraction should be performed from left to right.
$-\frac{}{2}-\frac{3}{}$

STEP 14

Perform the subtraction operation.
$-\frac{}{2}-3$

STEP 15

Convert the whole number to a fraction.
$-\frac{}{2}-\frac{}{2}$

STEP 16

Now, subtract the fractions.
$-\frac{}{2}$

STEP 17

Now, let's solve the fifth expression. According to BIDMAS, multiplication should be performed before addition.
$\frac{}{4}+\frac{3}{} \times \frac{}{3}$

STEP 18

Perform the multiplication operation.
$\frac{}{4}+\frac{}{8}$

STEP 19

Now, perform the addition operation.
$\frac{3}{8}$

STEP 20

Now, let's solve the sixth expression. According to BIDMAS, division should be performed before subtraction.
$\frac{5}{6} \div \frac{3}{4}-\frac{}{5}$

STEP 21

Perform the division operation.
$\frac{10}{9}-\frac{1}{5}$

STEP 22

Now, perform the subtraction operation.
$\frac{50}{45}-\frac{9}{45}$

STEP 23

Subtract the fractions.
$\frac{41}{45}$

STEP 24

Now, let's solve the seventh expression. According to BIDMAS, operations inside brackets should be performed first.
$\left(\frac{}{9}+\frac{8}{9}\right) \times \frac{1}{6}$

STEP 25

Perform the addition operation inside the brackets.
$1 \times \frac{1}{}$

STEP 26

Now, perform the multiplication operation.
$\frac{1}{6}$

STEP 27

Now, let's solve the eighth expression. According to BIDMAS, operations inside brackets should be performed first.
$\left(\frac{1}{}+\frac{}{5}\right) \times \frac{7}{}$

STEP 28

Perform the addition operation inside the brackets.
$\frac{19}{20} \times \frac{7}{8}$

STEP 29

Now, perform the multiplication operation.
$\frac{133}{160}$

STEP 30

Now, let's solve the ninth expression. According to BIDMAS, division should be performed before addition.
$\frac{2}{} \div \frac{8}{9}+\frac{5}{9}$

STEP 31

Perform the division operation.
$\frac{}{4}+\frac{5}{9}$

STEP 32

Now, perform the addition operation.
$\frac{27}{36}+\frac{20}{36}$

STEP 33

Add the fractions.
$\frac{47}{36}$ So, the solutions to the given expressions are $\frac{11}{24}, \frac{112}{243},2\frac{}{8}, -\frac{7}{2}, \frac{}{8}, \frac{41}{45}, \frac{1}{6}, \frac{133}{160}, \frac{47}{36}$

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STEP 1

Assumptions1. The operations must be performed in the correct order, following the BIDMAS/BODMAS rule (Brackets, Indices/Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). . The fractions are in simplest form.

STEP 2

Let's solve the first expression. According to BIDMAS, multiplication should be performed before addition.16+8×79\frac{1}{6}+\frac{}{8} \times \frac{7}{9}61​+8​×97​

STEP 3

Perform the multiplication operation.16+2172\frac{1}{6}+\frac{21}{72}61​+7221​

STEP 4

Now, perform the addition operation.16+724\frac{1}{6}+\frac{7}{24}61​+247​

STEP 5

To add these fractions, we need to find a common denominator. The least common multiple of and24 is24.424+724\frac{4}{24}+\frac{7}{24}244​+247​

STEP 6

Now, add the fractions.1124\frac{11}{24}2411​

STEP 7

Now, let's solve the second expression. According to BIDMAS, operations inside brackets should be performed first.9×(13+59)\frac{}{9} \times\left(\frac{1}{3}+\frac{5}{9}\right)9​×(31​+95​)

STEP 8

Perform the addition operation inside the brackets.8×1427\frac{8}{} \times \frac{14}{27}8​×2714​

STEP 9

Now, perform the multiplication operation.112243\frac{112}{243}243112​

STEP 10

Now, let's solve the third expression. According to BIDMAS, division should be performed before addition.89÷49+38\frac{8}{9} \div \frac{4}{9}+\frac{3}{8}98​÷94​+83​

STEP 11

Perform the division operation.+38+\frac{3}{8}+83​

STEP 12

Now, perform the addition operation.282\frac{}{8}28​

STEP 13

Now, let's solve the fourth expression. According to BIDMAS, subtraction should be performed from left to right.−2−3-\frac{}{2}-\frac{3}{}−2​−3​

STEP 14

Perform the subtraction operation.−2−3-\frac{}{2}-3−2​−3

STEP 15

Convert the whole number to a fraction.−2−2-\frac{}{2}-\frac{}{2}−2​−2​

STEP 16

Now, subtract the fractions.−2-\frac{}{2}−2​

STEP 17

Now, let's solve the fifth expression. According to BIDMAS, multiplication should be performed before addition.4+3×3\frac{}{4}+\frac{3}{} \times \frac{}{3}4​+3​×3​

STEP 18

Perform the multiplication operation.4+8\frac{}{4}+\frac{}{8}4​+8​

STEP 19

Now, perform the addition operation.38\frac{3}{8}83​

STEP 20

Now, let's solve the sixth expression. According to BIDMAS, division should be performed before subtraction.56÷34−5\frac{5}{6} \div \frac{3}{4}-\frac{}{5}65​÷43​−5​

STEP 21

Perform the division operation.109−15\frac{10}{9}-\frac{1}{5}910​−51​

STEP 22

Now, perform the subtraction operation.5045−945\frac{50}{45}-\frac{9}{45}4550​−459​

STEP 23

Subtract the fractions.4145\frac{41}{45}4541​

STEP 24

Now, let's solve the seventh expression. According to BIDMAS, operations inside brackets should be performed first.(9+89)×16\left(\frac{}{9}+\frac{8}{9}\right) \times \frac{1}{6}(9​+98​)×61​

STEP 25

Perform the addition operation inside the brackets.1×11 \times \frac{1}{}1×1​

STEP 26

Now, perform the multiplication operation.16\frac{1}{6}61​

STEP 27

Now, let's solve the eighth expression. According to BIDMAS, operations inside brackets should be performed first.(1+5)×7\left(\frac{1}{}+\frac{}{5}\right) \times \frac{7}{}(1​+5​)×7​

STEP 28

Perform the addition operation inside the brackets.1920×78\frac{19}{20} \times \frac{7}{8}2019​×87​

STEP 29

Now, perform the multiplication operation.133160\frac{133}{160}160133​

STEP 30

Now, let's solve the ninth expression. According to BIDMAS, division should be performed before addition.2÷89+59\frac{2}{} \div \frac{8}{9}+\frac{5}{9}2​÷98​+95​

STEP 31

Perform the division operation.4+59\frac{}{4}+\frac{5}{9}4​+95​

STEP 32

Now, perform the addition operation.2736+2036\frac{27}{36}+\frac{20}{36}3627​+3620​

STEP 33

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Let's solve the first expression. According to BIDMAS, multiplication should be performed before addition.
$\frac{1}{6}+\frac{}{8} \times \frac{7}{9}$

Perform the multiplication operation.
$\frac{1}{6}+\frac{21}{72}$

Now, perform the addition operation.
$\frac{1}{6}+\frac{7}{24}$

To add these fractions, we need to find a common denominator. The least common multiple of and24 is24.
$\frac{4}{24}+\frac{7}{24}$

Now, add the fractions.
$\frac{11}{24}$

Now, let's solve the second expression. According to BIDMAS, operations inside brackets should be performed first.
$\frac{}{9} \times\left(\frac{1}{3}+\frac{5}{9}\right)$

Perform the addition operation inside the brackets.
$\frac{8}{} \times \frac{14}{27}$

Now, perform the multiplication operation.
$\frac{112}{243}$

Now, let's solve the third expression. According to BIDMAS, division should be performed before addition.
$\frac{8}{9} \div \frac{4}{9}+\frac{3}{8}$

Perform the division operation.
$+\frac{3}{8}$

Now, perform the addition operation.
$2\frac{}{8}$

Now, let's solve the fourth expression. According to BIDMAS, subtraction should be performed from left to right.
$-\frac{}{2}-\frac{3}{}$

Perform the subtraction operation.
$-\frac{}{2}-3$

Convert the whole number to a fraction.
$-\frac{}{2}-\frac{}{2}$

Now, subtract the fractions.
$-\frac{}{2}$

Now, let's solve the fifth expression. According to BIDMAS, multiplication should be performed before addition.
$\frac{}{4}+\frac{3}{} \times \frac{}{3}$

Perform the multiplication operation.
$\frac{}{4}+\frac{}{8}$

Now, perform the addition operation.
$\frac{3}{8}$

Now, let's solve the sixth expression. According to BIDMAS, division should be performed before subtraction.
$\frac{5}{6} \div \frac{3}{4}-\frac{}{5}$

Perform the division operation.
$\frac{10}{9}-\frac{1}{5}$

Now, perform the subtraction operation.
$\frac{50}{45}-\frac{9}{45}$

Subtract the fractions.
$\frac{41}{45}$

Now, let's solve the seventh expression. According to BIDMAS, operations inside brackets should be performed first.
$\left(\frac{}{9}+\frac{8}{9}\right) \times \frac{1}{6}$

Perform the addition operation inside the brackets.
$1 \times \frac{1}{}$

Now, perform the multiplication operation.
$\frac{1}{6}$

Now, let's solve the eighth expression. According to BIDMAS, operations inside brackets should be performed first.
$\left(\frac{1}{}+\frac{}{5}\right) \times \frac{7}{}$

Perform the addition operation inside the brackets.
$\frac{19}{20} \times \frac{7}{8}$

Now, perform the multiplication operation.
$\frac{133}{160}$

Now, let's solve the ninth expression. According to BIDMAS, division should be performed before addition.
$\frac{2}{} \div \frac{8}{9}+\frac{5}{9}$

Perform the division operation.
$\frac{}{4}+\frac{5}{9}$

Now, perform the addition operation.
$\frac{27}{36}+\frac{20}{36}$