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 `=5x\xx10=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 × Number of Re 1 coins) + (Rs 2 × Number of Rs 2 coins) + (Rs 5 × 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