Question

QUESTION:-

Evaluate:

`\rm{}a. \displaystyle{ (3)/(4) + (5)/(6) + (7)/(8) \: \: \: \: \: \: \: \: \: \: \: \: b.6 (1)/(3) + 2 (5)/(6) + 3 (7)/(9) }`
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Answer 1

#a

`\large \sf (3)/(4) + (5)/(6) + (7)/(8)`

Write all numerators above the least common denominator 24

`\large \sf (3 * 6 +5 * 4 + 7 * 3 )/(24)`

Multiply the numbers

`\large \sf (18 +20 + 21 )/(24)`

Add the numbers

`\large \sf (59 )/(24)`

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#b

`\large \sf 6(1)/(3) + 2 (5)/(6) + 3 (7)/(9)`

Convert the mixed number to an improper fraction

`\large \sf (19)/(3) + (17)/(6) + (34)/(9)`

Write all numerators above the least common denominator 18

`\large \sf (19 * 6 + 17 * 3 + 34 * 2)/(18)`

Multiply the numbers

`\large \sf (114+ 51 + 68)/(18)`

Add the numbers

`\large \sf (233)/(18)`

Answer 2

#a

`\implies\displaystyle{ (3)/(4) + (5)/(6) + (7)/(8)}`

• LCM of 4,6 and 8=2×2×3×2=24

Converting each of them into equivalent like fractions,

`\implies{ \displaystyle (3)/(4) = (3 * 6)/(4 * 6) = (18)/(24) }`

`\implies{ \displaystyle{ (5)/(6) = (5 * 4)/(6 * 4) = (20)/(24) }}`

`\implies{ \displaystyle{ (7)/(8) = (7 * 3)/(8 * 3) = (21)/(24) }}`

`{ \displaystyle{ (3)/(4) + (5)/(6) + (7)/(8) = (18)/(24) + (20)/(24) + (21)/(24) = (59)/(24) = 2(11)/(24) }}`

Shortcut method:-

`\implies{ \displaystyle{ (3)/(4) + (5)/(6) + (7)/(8) }}`

`\implies{ \displaystyle{ (3 * 6 + 5 * 4 + 7 * 3)/(24) }}`

`\implies{ \displaystyle{ (18 + 20 + 21)/(24) }}`

`\implies{ \displaystyle{ (59)/(24) = 2 (11)/(24) }}`

Remarks:-

• Whenever the answer is an improper fractions we should convert the answer into mixed number.
• Always reduce the answer to the lowest terms.
• You should try to do mental work and apply the shortcut method.

#b

`\implies{ \displaystyle{6 (1)/(3) + 2 (5)/(6) + 3 (7)/(9) }}`

• Convert to improper fractions,

`\implies{ \displaystyle{ (19)/(3) + (17)/(6) + (34)/(9) }}`

• LCM of 3,6,9=3×3×2=18

`{ \displaystyle{ (19)/(3) + (17)/(6) + (34)/(9) = (19 * 6 + 17 * 3 + 34 * 2)/(18) }}`

`\implies{ \displaystyle{ (144 + 51 + 68)/(18) }}`

`\implies{ \displaystyle{ (233)/(18) = 12 (17)/(18) }}`

Alternatively,

`\implies{ \displaystyle{6 (1)/(3) + 2 (5)/(6) + 3 (7)/(9) }}`

`\implies{ \displaystyle{(6 + 2 + 3) + (1)/(3) + (5)/(6) + (7)/(9) }}`

`\implies{ \displaystyle{11 + (1 * 6 + 5 * 3 + 7 * 2)/(18) }}`

`\implies{ \displaystyle{11 + (35)/(18) }}`

`\implies{ \displaystyle{11 + 1 (17)/(18) }}`

`\implies{ \displaystyle{12 + (17)/(18) }}`

`\implies{ \displaystyle{12 (17)/(18) }}`