[4 marks sum] - MATHS STD 6 Questions - Vidyadip

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[4 marks sum]

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Question 14 Marks

The difference between the two numbers is 23. Taking the larger number as x, find:

(i) the expression for a smaller number.

(ii) the smaller number, if the sum of these two numbers is 91.

Answer

Difference of two numbers = 23
Let Larger number = x
(i) Then smaller number = x - 23
(ii) ∵ Sum of two numbers = 91
Then x + x - 23 = 91
⇒ 2x - 23 = 91
⇒ 2x = 91 + 23 = 114
$\Rightarrow x=\frac{114}{2}=57$
∴ Smaller number = x - 23 = 57 - 23 = 34

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Question 24 Marks

The difference between two numbers is 15. Taking the smaller number as x; find:

(i) the expression for the larger number.

(ii) the larger number, if the sum of these numbers is 71.

Answer

Difference of two numbers = 15
Let smaller number = x
∴ Second number = x + 15
∴ Larger number = x + 15
If sum of two numbers = 71
Then x + x + 15 = 7
(i) 2x + 15 = 71
⇒ 2x = 71 - 15 = 56
$x=\frac{56}{2}=28$
(ii) Larger number = x + 15 = 28 + 15 = 43

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Question 34 Marks

Two natural numbers differ by 6 and their sum is 36. Find the larger number.

Answer

∵ Difference between two numbers = 6
and their sum = 36
Let first natural number = x
The second number = x - 6
∴ x + x - 6 = 36
⇒ 2x = 36 + 6 = 42
$x=\frac{42}{2}=21$
∴ Larger number = 21

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Question 44 Marks

The difference between the ages of a woman and her son is 19 years and the sum of their ages is 37 years; find the age of the son.

Answer

Let age of son = x years
The age of woman = x + 19
∴ x + x + 19 = 37
⇒ 2x + 19 = 37
⇒ 2x = 37 - 19 = 18
$\Rightarrow x=\frac{18}{2}=9$
∴ Age of son = 9 years

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Question 54 Marks

The age of a man and the age of his daughter differ by 23 years and the sum of their ages is 41 years. Find the age of the man.

Answer

Let age of daughter = x years
Then age of man = (x + 23)
∴ x + (x + 23) = 41
x + x + 23 = 41
⇒ 2x + 23 = 41
⇒ 2x = 41 - 23 = 18
$\Rightarrow x=\frac{18}{2}=9$
∴ Age of man = x + 23 = 9 + 23 = 32 years

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Question 64 Marks

A whole number is increased by 7 and the new number so obtained is multiplied by 5; the result is 45. Find the whole number.

Answer

Let the required whole number = x
Then (x + 7) × 5 = 45
$\Rightarrow \frac{( x +7) \times 5}{5}=\frac{45}{5} \quad$ (Dividing by 5)
$\Rightarrow x+7=9$
$\Rightarrow x=9-7$
$x=2$
$\therefore$ Required whole number $=2$

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Question 74 Marks

One$-$fourth of a number added to one$-$sixth of It is $15.$ Find the number.

Answer

Let the required number $= x$
The $\frac{x}{4}+\frac{x}{6}=15$
$=\frac{3 x+2 x}{12}=15$
$\Rightarrow \frac{5 x}{12}=15$
$\Rightarrow x=\frac{15 \times 12}{5}$
$\Rightarrow x=36$
$\therefore$ Required number $=36$

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Question 84 Marks

Solve the following equation $: 2(z - 5) +3 (z + 2) - (3 - 5z) =10$

Answer

$2(z - 5) +3 (z + 2) - (3 - 5z) =10$
$\Rightarrow 2z - 10 + 3z + 6 - 3 + 5z = 10$
$\Rightarrow 10z - 7 = 10$
$\Rightarrow 10z - 7 + 7 = 10 + 7 ...($ Adding $7$ to both sides$)$
$\Rightarrow 10z = 17$
$\Rightarrow \frac{10 z }{10}=\frac{17}{10} ($ Dividing by $7)$
$\Rightarrow z =\frac{17}{10}=1 \frac{7}{10}$

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Question 94 Marks

Solve the following equation: 2 (x - 3) – 3 (x - 4) =12

Answer

2 (x - 3) – 3 (x - 4) =12

⇒ 2x - 6 - 3x + 12 = 12

⇒ - x + 6 = 12

⇒ - x + 6 - 6 = 12 - 6 ...(Subtracting 6 from both sides)

⇒ - x = 6

⇒ x = - 6

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Question 104 Marks

Solve the following equation: 7z + 15 = 3z – 13

Answer

7z + 15 = 3z – 13
⇒ 7z + 15 - 3z = 3z - 13 - 3z (Subtracting 3z from both sides)
⇒ 4z + 15 = - 13
⇒ 4z + 15 - 15 = - 13 - 15 ...(Subtracting 15 from both sides)
⇒ 4z = - 28
$\Rightarrow \frac{4 z }{4}=\frac{-28}{4} \quad \ldots($ Dividing by 4$)$
⇒ z = - 7

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Question 114 Marks

Solve the following equation: 5y – 15 = 27 -2y

Answer

5y – 15 = 27 -2y
⇒ 5y + 2y - 15 = 27 - 2y + 2y (Adding 2y to both sides)
⇒ 7y - 15 = 27
⇒ 7y - 15 + 15 = 27 + 15 ...(Adding 15 to both sides)
⇒ 7y = 42
$\Rightarrow \frac{7 y}{7}=\frac{42}{7} \quad \ldots($ Dividing by 7$)$
⇒ y = 6

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Question 124 Marks

Solve the following equation : $2 z -2 \frac{1}{2}=3 \frac{1}{3}$

Answer

$2 z -2 \frac{1}{2}=3 \frac{1}{3}$
$\Rightarrow 2 z -2 \frac{1}{2}+2 \frac{1}{2}=3 \frac{1}{3}+2 \frac{1}{2}..($ Adding $2 \frac{1}{2}$ to both sides$)$
$\Rightarrow 2 z=\frac{10}{3}+\frac{5}{2}$
$\Rightarrow \frac{20+15}{6}=\frac{35}{6}$
$\Rightarrow \frac{2 z }{2}=\frac{35}{6 \times 2} ....($ Dividing by $2)$
$\Rightarrow z =\frac{35}{12}=2 \frac{11}{12}$

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Question 134 Marks

Solve the following equation: 3z – 18 = z – (12 - 4z)

Answer

3z – 18 = z – (12 - 4z)
⇒ 3z - 18 = z - 12 + 4z
⇒ 3z - 18 = 5z - 12
⇒ 3z - 18 + 18 = 5z - 12 + 18 ...(Adding 18 to both sides)
⇒ 3z = 5z + 6
⇒ 3z - 5z = 5z + 6 - 5z (Subtracting 5z from both sides)
⇒ - 2z = 6
$\Rightarrow \frac{-2 z}{-2}=\frac{6}{-2} \ldots$ (Dividing by -2$)$
∴ z = - 3

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Question 144 Marks

Solve the following equation: 3y - (y + 2) = 4

Answer

3y - (y + 2) = 4
⇒ 3y - y - 2 = 4
⇒ 2y - 2 = 4
⇒ 2y - 2 + 2 = 4 + 2 ...(Adding 2 to both sides)
⇒ 2y = 6
$\Rightarrow \frac{2 y}{2}=\frac{6}{2} \ldots($ Dividing by 2$)$
∴ y = 3

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Question 154 Marks

Solve the following equation: 7x - 2 = 4x + 7

Answer

7x - 2 = 4x + 7
⇒ 7x - 2 + 2 = 4x + 7 + 2 ..(Adding 2 to both sides)
⇒ 7x = 4x + 9
⇒ 7x - 4x = 4x + 9 - 4x ...(Substracting 4x from both sides)
⇒ 3x = 9
$\Rightarrow \frac{3 x }{3}=\frac{9}{3}$...(Dividing by 3$)$
∴ x = 3

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Question 164 Marks

Solve the following equation $: 2x \div 3 = 7$

Answer

$2 x \div 3=7$
$\Rightarrow \frac{2 x }{3}=7$
$\Rightarrow \frac{2 x }{3} \times 3=7 \times 3 \ldots($ Multiplying by $3)$
$\Rightarrow 2 x =21$
$\Rightarrow \frac{2 x }{2}=\frac{21}{2} \ldots .($ Dividing by $2)$
$\Rightarrow x =\frac{21}{2}=10 \frac{1}{2}$

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Question 174 Marks

When two consecutive natural numbers are added, the sum is 31; find the numbers.

Answer

Let first natural number = x
Then second natural number = x + 1
According to the sum,
x + x + 1 = 31
⇒ 2x + 1 = 31
⇒ 2x = 31 - 1 = 30
$\Rightarrow x=\frac{30}{2}=15$
∴ First natural number = 15
and second number = 15 + 1 = 16

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Question 184 Marks

The difference between the ages of Gopal and his father is 26 years. If the sum of their ages is 56 years, find the ages of Gopal and his father.

Answer

Let age of Gopal = x years
∴ Age of his father = (x + 26) years
According to the sum,
x + x + 26 = 56
⇒ 2x + 26 = 56
⇒ 2x = 56 - 26 = 30
$\Rightarrow x=\frac{30}{2}=15$
∴ Age of Gopal = 15 years
and age of his father = 15 + 26 = 41 years

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Question 194 Marks

The age of a man is 27 years more than the age of his son. If the sum of their ages is 47 years, find the age of the son and his father.

Answer

Let the age of son = x years
∴ Age of his father = x + 27
According to the sum:
x + x + 27 = 47
⇒ 2x + 27 = 47
⇒ 2x = 47 - 27 = 20
$\Rightarrow x=\frac{20}{2}=10$
∴ Age of son = 10 years
and age of his father = 10 + 27 = 37 years

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Question 204 Marks

A number is increased by 12 and the new number obtained is multiplied by 5. If the resulting number is 95, find the original number.

Answer

Let the original number = x
According to the sum,
(x + 26) ÷3 = 18
$\Rightarrow \frac{x+26}{3}=18$
⇒ x + 26 = 18 × 3
⇒ x + 26 = 54
⇒ x = 54 - 26 = 28

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Question 214 Marks

A number is increased by 12 and the new number obtained is multiplied by 5. If the resulting number is 95, find the original number.

Answer

Let the original number = x
According to the sum,
(x + 12) × 5 = 95
⇒ 5x + 60 = 95
⇒ 5x = 95 - 60
⇒ 5x = 35
$\Rightarrow x=\frac{35}{5}=7$
∴ The original number = 7

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Question 224 Marks

Two$-$fifths of a number subtracted from three$-$fourths of it gives $56,$ find the number.

Answer

Let the required number $= x$
According to the sum,
$\frac{3}{4} x -\frac{2}{5} x =56$
$\frac{15 x -8 x }{20}=56$
$\Rightarrow \frac{7}{20} x =56$
$\Rightarrow x =\frac{56 \times 20}{7}=8 \times 20=160$
$\therefore$ Required number$ = 160$

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Question 234 Marks

One$-$fourth of a number added to two$-$seventh of it gives $135;$ find the number.

Answer

Let required number $= x$
$\therefore$ According to the sum,
$\frac{ x }{4}+\frac{2}{7} x =135$
$\Rightarrow \frac{7 x +8 x }{28}=135 \ldots( \text{LCM}$ of $4,7=28)$
$\Rightarrow \frac{15 x }{28}=135$
$\Rightarrow x =\frac{135 \times 28}{15}$
$\Rightarrow x = 9 \times 28 = 252$
$\therefore$ Required number $= 252$

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Question 244 Marks

The sum of three consecutive numbers is 54. Taking the middle number as x, find:

(i) expression for the smallest number and the largest number.

(ii) the three numbers.

Answer

Sum of three consecutive numbers = 54
Middle number = x
(i) The first number = x – 1
and third number = x + 1
(ii) ∴x + x-1+x+1 = 54
⇒ 3x = 54
$\Rightarrow x=\frac{54}{3}=18$
∴ First number =18 - 1 = 17
and third number =18 + 1 = 19
∴ Three required numbers are 17, 18,19

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Question 254 Marks

The difference between the two numbers is 23. Taking the larger number as x, find:

(i) the expression for a smaller number.

(ii) the smaller number, if the sum of these two numbers is 91.

Answer

Difference of two numbers = 23
Let Larger number = x
(i) Then smaller number = x - 23
(ii) ∵ Sum of two numbers = 91
Then x + x - 23 = 91
⇒ 2x - 23 = 91
⇒ 2x = 91 + 23 = 114
$\Rightarrow x=\frac{114}{2}=57$
∴ Smaller number = x - 23 = 57 - 23 = 34

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Question 264 Marks

The difference between two numbers is 15. Taking the smaller number as x; find:

(i) the expression for the larger number.

(ii) the larger number, if the sum of these numbers is 71.

Answer

Difference of two numbers = 15
Let smaller number = x
∴ Second number = x + 15
∴ Larger number = x + 15
If sum of two numbers = 71
Then x + x + 15 = 71
(i) 2x + 15 = 71
⇒ 2x = 71 - 15 = 56
$x=\frac{56}{2}=28$
(ii) Larger number = x + 15 = 28 + 15 = 43

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Question 274 Marks

Two natrual numbers differ by 6 and sum of them is 36. Find the larger number.

Answer

∵ Difference between two numbers = 6
and their sum = 36
Let first natural number = x
The second number = x - 6
∴ x + x - 6 = 36
⇒ 2x = 36 + 6 = 42
$x=\frac{42}{2}=21$
∴ Larger number = 21

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Question 284 Marks

The difference between the ages of a woman and her son is 19 years and the sum of their ages is 37 years; find the age of the son.

Answer

Let age of son = x years
The age of woman = x + 19
∴ x + x + 19 = 37
⇒ 2x + 19 = 37
⇒ 2x = 37 - 19 = 18
$\Rightarrow x=\frac{18}{2}=9$
∴ Age of son = 9 years

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Question 294 Marks

The age of a man and the age of his daughter differ by 23 years and the sum of their ages is 41 years. Find the age of the man.

Answer

Let age of daughter = x years
Then age of man = (x + 23)
∴ x + (x + 23) = 41
x + x + 23 = 41
⇒ 2x + 23 = 41
⇒ 2x = 41 - 23 = 18
$\Rightarrow x=\frac{18}{2}=9$
∴ Age of man = x + 23 = 9 + 23 = 32 years

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Question 304 Marks

Solve : $5 \frac{2}{3}- z =2 \frac{1}{2}$

Answer

$5 \frac{2}{3}- z =2 \frac{1}{2}$
$\Rightarrow \frac{17}{3}- z =\frac{5}{2}$
$\Rightarrow \frac{17}{3}-\frac{5}{2}= z$
$\Rightarrow 6 \times z =\frac{17}{3} \times 6-\frac{5}{2} \times 6($ Multiply both side by $6 )$
$\Rightarrow 6 z =34-15$
$\Rightarrow 6 z =19$
$\Rightarrow z =\frac{19}{6}$
$\Rightarrow z =3 \frac{1}{6}$

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Question 314 Marks

Solve : $y-3 \frac{1}{2}=2 \frac{1}{3}$

Answer

$y -3 \frac{1}{2}=2 \frac{1}{3}$
$\Rightarrow y -\frac{7}{2}=\frac{7}{3}$
$\Rightarrow y =\frac{7}{3}+\frac{7}{2}$
$\Rightarrow 6 \times y=\frac{7}{3} \times 6+\frac{7}{2} \times 6 \quad ($ Multiply both side by $6)$
$\Rightarrow 6 y=14+21$
$\Rightarrow y=\frac{35}{6}$
$\Rightarrow y=5 \frac{5}{6}$

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Question 324 Marks

Solve: 3(2x + 1) -2(x - 5) -5(5 - 2x) = 16

Answer

3(2x + 1) -2(x - 5) -5(5 - 2x) = 16
⇒ 6x + 3 - 2x + 10 - 25 + 10x = 16
⇒ 6x - 2x + 10x + 3 + 10 - 25 = 16
⇒ 16x - 2x + 13 - 25 = 16
⇒ 14x - 12 = 16
⇒ 14x = 16 + 12 = 28
$\Rightarrow x=\frac{28}{14}=2$
∴ x = 2

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Question 334 Marks

Solve: 5(x - 2)- 2(x + 2) = 3

Answer

5(x - 2)- 2(x + 2) = 3
⇒ 5x - 10 - 2x - 2 = 3
⇒ 5x - 2x - 10 - 2 = 3
⇒ 3x - 12 = 3
⇒ 3x = 3 + 12
$\Rightarrow x=\frac{15}{3}=5$
∴ x = 5

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Question 344 Marks

Solve $: 3x + 2 = -2.2$

Answer

$3x + 2 = -2.2$
$\Rightarrow 3x = - 2.2 - 2$
$\Rightarrow 3x = - 4.2$
$\Rightarrow x=\frac{-42}{3 \times 10}$
$\Rightarrow x=-\frac{14}{10}$
$\Rightarrow x=-\frac{7}{5}$
$\Rightarrow x=-1 \frac{2}{5}$
$\Rightarrow x = - 1.4$

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Question 354 Marks

Solve $: 5x - 14 = x - (24 + 4x)$

Answer

$5x - 14 = x - (24 + 4x)$
$\Rightarrow 5x - 14 = x - 24 - 4x$
$\Rightarrow 5x + 4x - x = - 24 + 14$
$\Rightarrow 9x - x = - 10$
$\Rightarrow 8x = - 10$
$\Rightarrow x=\frac{-10}{8}$
$\Rightarrow x=-\frac{5}{4}$ or $-1 \frac{1}{4}$

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Question 364 Marks

Solve $: - m + (3m - 6m) = - 8 - 14$

Answer

$-m+(3 m-6 m)=-8-14$
$\Rightarrow-m+3 m-6 m=-8-14$
$\Rightarrow-7 m+3 m=-22$
$\Rightarrow-4 m=-22$
$\Rightarrow m=\frac{-22}{-4}$
$\Rightarrow m=\frac{11}{2}$
$\Rightarrow m=5 \frac{1}{2}$

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Question 374 Marks

Solve: 8x + 6 + 2x - 4 = 4x + 8

Answer

8x + 6 + 2x - 4 = 4x + 8
⇒ 10x + 2 = 4x + 8
⇒ 10x - 4x = 8 - 2
⇒ 6x = 6
$\Rightarrow x=\frac{6}{6}$
⇒ x = 1

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Question 384 Marks

Solve $: 3.5x - 9 - 3 = x + 1$

Answer

$\Rightarrow 3.5x - x = 1 + 9 + 3$
$\Rightarrow 2.5x = 13$
$\Rightarrow x=\frac{13}{2.5}$
$\Rightarrow x=\frac{13 \times 10}{25}$
$\Rightarrow x=\frac{26}{5}$ or $5 \frac{1}{5}$

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Question 394 Marks

Solve: $\frac{ z }{7}+1=2 \frac{1}{2}$

Answer

$\frac{ z }{7}+1=2 \frac{1}{2}$
$\Rightarrow \frac{ z }{7}+1=\frac{5}{2}$
$\Rightarrow \frac{ z }{7}=\frac{5}{2}-1$
$\Rightarrow \frac{ z }{7}=\frac{5-2}{2}$
$\Rightarrow \frac{ z }{7}=\frac{3}{2}$
$\Rightarrow z =\frac{3}{2} \times 7$
$\Rightarrow z =\frac{21}{2}$
$\Rightarrow z =10 \frac{1}{2}$

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Question 404 Marks

Solve: $-2 \frac{1}{5}=y-4$

Answer

$-2 \frac{1}{5}=y-4$
$\Rightarrow-\frac{11}{5}=y-4$
$\Rightarrow y-4=-\frac{11}{5}$
$\Rightarrow y=-\frac{11}{5}+4$
$\Rightarrow y=\frac{-11+20}{5}$
$\Rightarrow y=\frac{9}{5}$
$\Rightarrow y=1 \frac{4}{5}$

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Question 414 Marks

Solve : $5=m+3 \frac{4}{7}$

Answer

$5= m +3 \frac{4}{7}$
$\Rightarrow 5= m +\frac{25}{7}$
$\Rightarrow m +\frac{25}{7}=5$
$\Rightarrow m =5-\frac{25}{7}$
$\Rightarrow m =\frac{35-25}{7}$
$\Rightarrow m =\frac{10}{7}$
$\Rightarrow m =1 \frac{3}{7}$

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Question 424 Marks

Solve : $3 \frac{2}{5}= x -2 \frac{1}{3}$

Answer

$3 \frac{2}{5}= x -2 \frac{1}{3}$
$\Rightarrow \frac{17}{5}= x -\frac{7}{3}$
$\Rightarrow x -\frac{7}{3}=\frac{17}{5}$
$\Rightarrow x =\frac{17}{5}+\frac{7}{3}$
$\Rightarrow x =\frac{51+35}{15}$
$\Rightarrow x =\frac{86}{15}$
$\Rightarrow x=5 \frac{11}{15}$

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[4 marks sum] - MATHS STD 6 Questions - Vidyadip