Linear Equations

Exercise 2.2 Part 3

Question 12: Fifteen years from now Ravi’s age will be four times his present age. What is Ravi’s present age?

Solution: Let the present age of Ravi = m year
Age of Ravi after 15 years = m + 15 year

According to question, age of Ravi will be four times of his present age.
i.e. Age of Ravi after 15 year = 4 x present age of Ravi
`⇒ m + 15 = 4 xx m`
`⇒ m + 15 = 4 m`

After transposing m to RHS, we get
`15 = 4m – 3`
`⇒ 15 = 3m`

After dividing both sides by 3, we get

`(15)/(3)=(3m)/(3)`

Or, `m=5`

Thus, Ravi’s present age = 5 year 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 `-(7)/(12)`. What is the number?

Solution: Let the rational number `=a/b`

As per question:

`(text(Rational number)xx5/2)+2/3=-(7)/(12)`

Or, `(a/bxx5/2)+2/3=-(7)/(12)`

By transposing `2/3` to RHS we get:

`(a/bxx5/2) =-(7)/(12) -2/3`

Or, `a/bxx5/2=(-7-8)/(12)`

Or, `a/bxx5/2=-(15)/(12)`

By dividing both sides by `5/2` we get:

`a/bxx5/2÷5/2=-(15)/(12)÷5/2`

Or, `a/bxx5/2xx2/5=-(15)/(12)xx2/5`

Or, `a/b=-3/6=-1/2`


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: Let the number of Rs. 100 notes `=2x`

Number of Rs. 50 notes `=3x`

Number of Rs. 10 notes `=5x`

So, value of Rs. 100 notes `=2x\xx100=200x`

Value of Rs. 50 notes `=3x\xx50=150x`

Value of Rs. 10 notes `=5xxx10=50x`

As per question, total cash `=4,00,000=200x+150x+50x`

Or, `400x=4,00,000`

By dividing both sides by 400 we get:

`(400x)/(400)=(4,00,000)/(400)`

Or, `x=1000`

Substituting the value of x we can find the number of notes of different denominations as follows:

Number of Rs. 100 notes `=2x=2xx1000=2000`

Number of Rs. 50 notes `=3x=3xx1000=3000`

Number of Rs. 10 notes `=5x=5xx1000=5000`

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
Total number of coins = 160
Coins of denomination = Re 1, Rs 2 and Rs 5
Number of Rs 2 coins = 3 x number Rs 5 coins

Let the number of coins of Rs 5 = m
Since, the number coins of Rs 2 is 3 times of the number of coins of Rs 5
Therefore, number of coins of Rs 2 `= m xx 3 = 3m`
Now, Number of coins of Re 1 = Total number of coins – (Number of Rs 5 coins + Number of Rs 2 coins)

Therefore,
Number of coins of Re 1 `= 160 – (m + 3m) = 160 – 4m`
Total Rs = (Re 1 x Number of Re 1 coins) + (Rs 2 x Number of Rs 2 coins) + (Rs 5 x Number of Rs 5 coins)
`⇒ 300 = [1 xx (160 – 4m)] + (2 xx 3m) + (5 xx m)`
`⇒ 300 = (160 – 4m) + 6m + 5m`
`⇒ 300 = 160 – 4m + 6m + 5m`
`⇒ 300 = 160 – 4m + 11m`
`⇒ 300 = 160 + 7m`

After transposing 160 to LHS, we get
`300 – 160 = 7m`
`⇒ 140 = 7 m`

After dividing both sides by 7, we get

`(140)/(7)=(7m)/(7)`

Or, `m=20`

Thus, number of coins of Rs 5 = 20
Now, since, number of coins of Re 1 `= 160 – 4m`
Thus, by substituting the value of m, we get
Number of coins of Re 1 `= 160 – (4 xx 20) = 160 – 80 = 80`
Now, number coins of Rs 2 = 3m

Thus, by substituting the value of m, we get
Number of coins of Rs 2 `= 3m = 3 xx 20 = 60`
Therefore,
Number of coins of Re 1 = 80
Number of coins of Rs 2 = 60
Number of coins of Rs 5 = 20

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

Solution: Given, Total number participants = 63
Total prize money distributed = Rs 3000
Winner gets a prize of Rs 100
Loser gets a prize of Rs 25
Number of winners = ?

Let the number of winners = m
Since,
Number of winners + Number of losers = Total number of participants
Or, m + Number of losers = 63

By transposing ‘m’ to RHS, we get
Number of losers `= 63 – m`
Now, Total Prize money distributed to winners
= Number of winners X prize money distributed to each winner `= m xx 100 = 100m`
Total prize money distributed to losers
= Number of losers X prize money distributed to each loser
`= (63 – m) xx 25 = (63 xx 25) – 25 m = 1575 – 25 m`

Now, Total Prize money of winners + Total Prize money of losers = Total prize money
By substituting the total prize money distributed to winners and total prize money distributed to losers, we get
`100 m + 1575 – 25 m = 3000`
`⇒ 100 m – 25 m + 1575 = 3000`

By transposing 1575 to RHS, we get
`100 m – 25 m = 3000 – 1575`
`⇒75 m = 1425`

After dividing both sides by 75, we get

`(75m)/(75)=(1425)/(75)`

Or, `m=19`

Thus, number of winners = 19 Answer



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