Linear Equations in One Variable - 8th math

Question (12) Fifteen years from now Ravi's age will be four times his present age. What is Ravi's present age?

Solution:

Given, Fifteen years from now, the age of ravi = present age of ravi × 4

Thus, present age of Ravi = ?

Let the present age of Ravi = a year

Therefore, After 15 years from now age of Ravi = a + 15 year

According to question, after 15 years now the age of Ravi = a × 4

Thus, Ravi present age + 15 = a × 4

⇒ a + 15 = a × 4

After rearranging the above expression, we get

⇒ 4 a = a + 15

After transposing a to LHS, we get

⇒ 4 a – a = 15

⇒ 3 a = 15

After transposing 3 to RHS, we get

⇒ a = 15/3 = 5

Thus, the present age of Ravi = 5 years Answer

Question (13) A rational number is such that when you multiply it by 5/2 and add 2/3 to the product, you get . What is the number?

Solution: Let the rational number = a/b

As per question:

By transposing 2/3 to RHS we get:

After transposing 5/2 to RHS, we get

Thus, required rational number = –1/2 Answer

Question (14) Lakshmi is a cashier in a bank. She has currency notes of denominations Rs 100, Rs 50 and Rs 10, respectively. The ratio of the number of these notes is 2:3:5. The total cash with Lakshmi is Rs 4,00,000. How many notes of each denomination does she have?

Solution:

Given, Currency notes with cashier = ₹ 100, ₹ 50 and ₹ 10

Ratio of currency notes = 2:3:5

Sum of cash = ₹ 400000

Thus, number of each denominations = ?

Since, currency notes in the denominations of ₹ 100, ₹ 50 and ₹ 10 are in the ratio of 2:3:5

Therefore, Let the number of ₹ 100 notes =2x

Thus, the value of notes of ₹ 100 = 2 × 100 = 200 x

And the number of ₹ 50 notes =3x

Thus, the value of notes of ₹ 20 = 3 × 50 = 150 x

And the number of ₹ 10 notes =5x

Thus, the value of notes of ₹ 10 = 5 × 10 = 50 x

Now, according to question, total cash = ₹ 400000

Thus, ₹ 200 x + ₹ 150 x + ₹ 50 x = ₹ 400000

⇒ ₹ 400 x = ₹ 400000

After transposing 400 to RHS, we get

x = ₹ 400000/₹ 400

⇒ x =1000

Thus, number of currency notes in the given denominations

Since, number of notes in the denomination of ₹ 100 = 2 x

Thus, number of ₹ 100 notes = 2 × 1000 = 2000

And, since, number of notes in the denomination of ₹ 50 = 3 x

Thus, number of ₹ 50 notes = 3 × 1000 = 3000

And, since, number of notes in the denomination of ₹ 10 = 5 x

Thus, number of ₹ 10 notes = 5 × 1000 = 5000

Thus, number of ₹ 100 notes = 2000, number of ₹ 50 notes = 3000 and number of ₹ 10 notes = 5000 Answer

Question (15) I have a total of Rs 300 in coins of denomination Re 1, Rs 2 and Rs 5. The number of Rs 2 coins is 3 times the number of Rs 5 coins. The total number of coins is 160. How many coins of each denomination are with me?

Solution:

Given, total value of Rs = Rs 300

Coins of denomination ₹ , ₹ 2 and ₹ 5

The total number of coins = 160

Number of ₹ 2 coins = 3 × number of ₹ 5 coins

Thus number of coins of each denominations = ?

Let the number of coins of ₹ 5 = a

Now, since number of ₹ 2 coins = 3 × number of ₹ 5 coins

Thus, number of coins of ₹ 2 = 3 × a = 3 a

Now, as given in the question, the total number of coins = 160

Thus, number of coins of ₹ 1 = Total number of coin – (Number of coins of ₹ 2 + Number of coins of ₹ 5)

Thus, number of coins of ₹ 1 = 160 – (3a + a)

⇒ number of coins of ₹ 1 = 160 – 4a

Now, as given in the question, the total values of rupees = ₹ 300

Thus, (₹ 1 × Number of ₹ 1 coins) + (₹ 2 × Number of ₹ 2 coins) + (₹ 5 × Number of ₹ 5 coins) = ₹ 300

⇒ [1 × (160 – 4 a)] + (2 × 3 a) + (5 × a) = ₹ 300

⇒ 160 – 4 a + 6 a + 5 a = ₹ 300

⇒ 160 – 4 a + 11 a = 300

⇒ 160 – 7 a = 300

After transposing 160 to RHS, we get

⇒ 7 a = 300 – 160

⇒ 7 a = 140

After transposing 7 to RHS, we get

a = 140/7

⇒ a = 20

Thus, since number of ₹ 1 coins = 160 – 4 a

Thus, after substituting the value of a = 20, we get

Number of ₹ 1 coins = 160 – (4 × 20)

= 160 – 80 = 80

Thus, number of ₹ 1 coins = 80

And, since number of ₹ 2 coins = 3 a

Thus, after substituting the value of a = 20, we get

Number of ₹ 1 coins = 3 × 20 = 60

And, since number of ₹ 5 coins = a

Thus, after substituting the value of a = 20, we get

Number of ₹ 5 coins = 20

Thus, number of coins of ₹ 1 = 80, number of coins of ₹ 2 = 60, and number of coins of ₹ 5 = 20 Answer

Question (16) The organisers of an essay competition decide that a winner in the competition gets a prize of ₹ 100 and a participant who does not win gets a prize of ₹ 25. The total prize money distributed is ₹ 3,000. Find the number of winners, if the total number of participants is 63.

Solution:

Given, Total number participants = 63

And, total prize money distributed = ₹ 3000

And, prize gets by winner = ₹ 100

And prize gets by those who does not win = ₹ 25

Thus, number of winners = ?

Let the total number of winners = w

Now, according to question, since the total number of participants = 63

Thus, number of participants who does not win = 63 – number of winners

And hence, number of participants who does not win = 63 – w

Now, according to question,

Number of winners × prize money to winners + Number of losers × prize money to losers = ₹ 3000

⇒ w × ₹ 100 + [(63 – w) × ₹ 25] = ₹ 3000

⇒ 100 w + (1575 – 25 w) = 3000

⇒ 100 w + 1575 – 25 w = 3000

After transposing 1575 to RHS, we get

⇒ 100 w – 25 w = 3000 – 1575

⇒ 75 w = 1425

After transposing 75 to RHS, we get

w = 1425/75

⇒ w = 19

Thus, number of winners = 19 Answer