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Answer 1

Questions

1. Of all the consecutive natural numbers, five times the smallest number is 9 more than four times the greatest number, find the numbers.

2. Raju sold a bicycle to Amit at 8% profit. Amit repaired it spending ₹54. Then he sold the bicycle to Nikhil for ₹1134 with no loss and no profit. Find the cost price of the bicycle of which Raju purchased it.

3. A cricket player scored 180 runs in the first match and 257 runs in the second match. Find the number of runs he should score in the third match so that the average of runs in the the three matches be 230.

4. Sudhir's present age is 5 more than three times the age of Viru. Anil's age is half the age of Sudhir. If the ratio of the sum of Sudhir's and Viru's age to three times Anil's age is 5:6, then find Viru's age.

_______________________________________

Answers

1. Let's consider the three consecutive numbers to be x-1, x and x+1.

Now we will have to solve this equation to find the each consecutive number ⇒  5(x - 1) = 9 + 4(x + 1)

Let's solve your equation step-by-step.

5(x - 1) = 9 + 4(x + 1)

Step 1: Simplify the equation.

⇒ 5(x - 1) = 9 + 4(x + 1)

⇒ 5(x) - 5(1) = 9 + 4(x) + 4(1)

⇒ 5x - 5 = 9 + 4x + 4

Step 2 : Combine Like terms.

⇒ 5x - 5 = 9 + 4x + 4

⇒ 5x - 5 = 9 + 4 + 4x

⇒ 5x - 5 = 13 + 4x

Step 3: Subtract 4x from both sides of the equation.

⇒ 5x - 5 - 4x = 13 + 4x - 4x

⇒ x - 5 = 13

Step 4: Add 5 to both sides of the equation.

⇒ x - 5 + 5 = 13 + 5

⇒ x = 18

∴ The 1st Consecutive Number is ⇒ x - 1 = 18 - 1 = 17

∴ The 2nd Consecutive Number is ⇒ x = 18

∴ The 3rd Consecutive Number is ⇒ x + 1 = 18 + 1 = 19

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2. Let's consider the price Raju purchased the bicycle to be 'x'.

Selling Price of the Bicycle is 108% of the cost price.

⇒ 108% of x

⇒  $\dfrac{108x}{100}$

Amount Amit repaired the bicycle at is ⇒ ₹54

Price Nikhil purchased the bicycle for ⇒  $\dfrac{108x}{100} + 54 = 1152$

Let's solve your equation step-by-step.

$\dfrac{108x}{100} + 54 = 1134$

Step 1: Subtract 54 from both sides of the equation.

⇒ $\dfrac{108x}{100} + 54 -54= 1134 - 54$

⇒ $\dfrac{108x}{100} = 1080$

Step 2: Cross multiply.

⇒  $\dfrac{108x}{100} = 1080$

⇒  $108x = 1080\times 100$

⇒ $108x = 108000$

Step 3: Divide 108 from both sides of the equation.

⇒  $\dfrac{108x}{108} = \dfrac{108000}{108}$

⇒ x = 100 0

∴ The cost price of the bicycle for which Raju purchased was ₹1000.

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3. Number of runs scored in first match ⇒ 180

Number of runs scored in second match ⇒ 257

Numbers of runs scored in third match ⇒ x

The average run given is 230.

Average =  $\sf \dfrac{Sum \ of \ Runs\ in\ all\ matches }{No.\ of \ matches}$

Let's solve your equation step-by-step.

$\dfrac{180+257+x}{3} =230$

Step 1: Simplify the equation.

⇒ $\dfrac{180+257+x}{3} =230$

⇒ $\dfrac{437+x}{3}=230$

Step 2: Cross multiply.

⇒ $\dfrac{437+x}{3}=230$

⇒  437 + x = 230 × 3

⇒  437 + x = 690

Step 3: Subtract 437 from both sides of the equation.

⇒  437 + x -437 = 690 - 437

⇒ x = 253

∴ The total number of runs scored by the cricketer in the third match was 253 runs.

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4. Let's consider Viru's present age to be 'x'.

Sudhir's Present Age would be ⇒ 3x + 5

Anil's Present Age would be ⇒ $\dfrac{3x+5}{2}$

Given is that the ratio of the sum of the age of Sudhir and Viru to three times Anil's age is 5:6.

We will solve this equation to find the age of Viru ⇒ $\dfrac{x+3x+5}{3(\frac{3x+5}{2} )} = \dfrac{5}{6}$

Let's solve your equation step-by-step.

$\dfrac{x+3x+5}{3(\frac{3x+5}{2} )} = \dfrac{5}{6}$

Step 1: Cross multiply.

⇒ $6(x+3x+5)=5[3(\frac{3x+5}{2})]$

⇒ $6(x)+6(3x)+6(5) =\dfrac{45}{2} x +\dfrac{75}{2}$

⇒ $6x+18x+30 =\dfrac{45}{2} x +\dfrac{75}{2}$

Step 2: Combine Like terms.

⇒ $(6x+18x)+30 =\dfrac{45}{2} x +\dfrac{75}{2}$

⇒ $24x+30 =\dfrac{45}{2} x +\dfrac{75}{2}$

Step 3: Subtract 30 from both sides of the equation.

⇒ $24x+30 -30=\dfrac{45}{2} x +\dfrac{75}{2} - 30$

⇒ $24x= \dfrac{45}{2} x +\dfrac{75}{2} - \dfrac{60}{2}$

⇒ $24x= \dfrac{45}{2} x +\dfrac{15}{2}$

Step 4: Subtract (⁴⁵/₂)x from both sides of the equation.

⇒ $24x - \dfrac{45}{2} x = \dfrac{45}{2} x - \dfrac{45}{2} x +\dfrac{15}{2}$

⇒ $\dfrac{48}{2} x - \dfrac{45}{2} x = +\dfrac{15}{2}$

⇒ $\dfrac{3}{2} x = \dfrac{15}{2}$

Step 5: Multiply ²/₃ from both sides of the equation.

⇒ $\dfrac{2}{3}\times \dfrac{3}{2} x =\dfrac{2}{3}\times \dfrac{15}{2}$

⇒ $x = 5$

∴ Viru is 5 years old.

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Answer 2

Answer:

Questions

1. Of all the consecutive natural numbers, five times the smallest number is 9 more than four times the greatest number, find the numbers.

2. Raju sold a bicycle to Amit at 8% profit. Amit repaired it spending ₹54. Then he sold the bicycle to Nikhil for ₹1134 with no loss and no profit. Find the cost price of the bicycle of which Raju purchased it.

3. A cricket player scored 180 runs in the first match and 257 runs in the second match. Find the number of runs he should score in the third match so that the average of runs in the the three matches be 230.

4. Sudhir's present age is 5 more than three times the age of Viru. Anil's age is half the age of Sudhir. If the ratio of the sum of Sudhir's and Viru's age to three times Anil's age is 5:6, then find Viru's age.

_______________________________________

Answers

1. Let's consider the three consecutive numbers to be x-1, x and x+1.

Now we will have to solve this equation to find the each consecutive number ⇒ 5(x - 1) = 9 + 4(x + 1)

Let's solve your equation step-by-step.

5(x - 1) = 9 + 4(x + 1)

Step 1: Simplify the equation.

⇒ 5(x - 1) = 9 + 4(x + 1)

⇒ 5(x) - 5(1) = 9 + 4(x) + 4(1)

⇒ 5x - 5 = 9 + 4x + 4

Step 2 : Combine Like terms.

⇒ 5x - 5 = 9 + 4x + 4

⇒ 5x - 5 = 9 + 4 + 4x

⇒ 5x - 5 = 13 + 4x

Step 3: Subtract 4x from both sides of the equation.

⇒ 5x - 5 - 4x = 13 + 4x - 4x

⇒ x - 5 = 13

Step 4: Add 5 to both sides of the equation.

⇒ x - 5 + 5 = 13 + 5

⇒ x = 18

∴ The 1st Consecutive Number is ⇒ x - 1 = 18 - 1 = 17

∴ The 2nd Consecutive Number is ⇒ x = 18

∴ The 3rd Consecutive Number is ⇒ x + 1 = 18 + 1 = 19

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2. Let's consider the price Raju purchased the bicycle to be 'x'.

Selling Price of the Bicycle is 108% of the cost price.

⇒ 108% of x

⇒ \dfrac{108x}{100}

100

108x

Amount Amit repaired the bicycle at is ⇒ ₹54

Price Nikhil purchased the bicycle for ⇒ \dfrac{108x}{100} + 54 = 1152

100

108x

+54=1152

Let's solve your equation step-by-step.

\dfrac{108x}{100} + 54 = 1134

100

108x

+54=1134

Step 1: Subtract 54 from both sides of the equation.

⇒ \dfrac{108x}{100} + 54 -54= 1134 - 54

100

108x

+54−54=1134−54

⇒ \dfrac{108x}{100} = 1080

100

108x

=1080

Step 2: Cross multiply.

⇒ \dfrac{108x}{100} = 1080

100

108x

=1080

⇒ 108x = 1080\times 100108x=1080×100

⇒ 108x = 108000108x=108000

Step 3: Divide 108 from both sides of the equation.

⇒ \dfrac{108x}{108} = \dfrac{108000}{108}

108

108x

=

108

108000

⇒ x = 100 0

∴ The cost price of the bicycle for which Raju purchased was ₹1000.

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3. Number of runs scored in first match ⇒ 180

Number of runs scored in second match ⇒ 257

Numbers of runs scored in third match ⇒ x

The average run given is 230.

Average = \sf \dfrac{Sum \ of \ Runs\ in\ all\ matches }{No.\ of \ matches}

No. of matches

Sum of Runs in all matches

Let's solve your equation step-by-step.

\dfrac{180+257+x}{3} =230

3

180+257+x

=230

Step 1: Simplify the equation.

⇒ \dfrac{180+257+x}{3} =230

3

180+257+x

=230

⇒ \dfrac{437+x}{3}=230

3

437+x

=230

Step 2: Cross multiply.

⇒ \dfrac{437+x}{3}=230

3

437+x

=230

⇒ 437 + x = 230 × 3

⇒ 437 + x = 690

Step 3: Subtract 437 from both sides of the equation.

⇒ 437 + x -437 = 690 - 437

⇒ x = 253

∴ The total number of runs scored by the cricketer in the third match was 253 runs.

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4. Let's consider Viru's present age to be 'x'.

Sudhir's Present Age would be ⇒ 3x + 5

Anil's Present Age would be ⇒ \dfrac{3x+5}{2}

2

3x+5

Given is that the ratio of the sum of the age of Sudhir and Viru to three times Anil's age is 5:6.

We will solve this equation to find the age of Viru ⇒ \dfrac{x+3x+5}{3(\frac{3x+5}{2} )} = \dfrac{5}{6}

3(

2

3x+5

)

x+3x+5

=

6

5

Let's solve your equation step-by-step.

\dfrac{x+3x+5}{3(\frac{3x+5}{2} )} = \dfrac{5}{6}

3(

2

3x+5

)

x+3x+5

=

6

5

Step 1: Cross multiply.

⇒ 6(x+3x+5)=5[3(\frac{3x+5}{2})]6(x+3x+5)=5[3(

2

3x+5

)]

⇒ 6(x)+6(3x)+6(5) =\dfrac{45}{2} x +\dfrac{75}{2}6(x)+6(3x)+6(5)=

2

45

x+

2

75

⇒ 6x+18x+30 =\dfrac{45}{2} x +\dfrac{75}{2}6x+18x+30=

2

45

x+

2

75

Step 2: Combine Like terms.

⇒ (6x+18x)+30 =\dfrac{45}{2} x +\dfrac{75}{2}(6x+18x)+30=

2

45

x+

2

75

⇒ 24x+30 =\dfrac{45}{2} x +\dfrac{75}{2}24x+30=

2

45

x+

2

75

Step 3: Subtract 30 from both sides of the equation.

⇒ 24x+30 -30=\dfrac{45}{2} x +\dfrac{75}{2} - 3024x+30−30=

2

45

x+

2

75

−30

⇒ 24x= \dfrac{45}{2} x +\dfrac{75}{2} - \dfrac{60}{2}24x=

2

45

x+

2

75

−

2

60

⇒ 24x= \dfrac{45}{2} x +\dfrac{15}{2}24x=

2

45

x+

2

15

Step 4: Subtract (⁴⁵/₂)x from both sides of the equation.

⇒ 24x - \dfrac{45}{2} x = \dfrac{45}{2} x - \dfrac{45}{2} x +\dfrac{15}{2}24x−

2

45

x=

2

45

x−

2

45

x+

2

15

⇒ \dfrac{48}{2} x - \dfrac{45}{2} x = +\dfrac{15}{2}

2

48

x−

2

45

x=+

2

15

⇒ \dfrac{3}{2} x = \dfrac{15}{2}

2

3

x=

2

15

Step 5: Multiply ²/₃ from both sides of the equation.

⇒ \dfrac{2}{3}\times \dfrac{3}{2} x =\dfrac{2}{3}\times \dfrac{15}{2}

3

2

×

2

3

x=

3

2

×

2

15

⇒ x = 5x=5

∴ Viru is 5 years old.

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