Inequalities in One Unknown

Q:There are 100 coins in a bag. n of them are $5 coins and the rest are$2 coins.It is known that the total amount of these coins is greater than 300.
(a) Find the least value of n.
(b)If the number of $5 coins is doubled and the number of $2 coins is halved, determinewhether the total amount of these coins will exceed $400 or not. Explain youranswer.


(a)
No. of $2 coins = 100 - n

Total amount of these coins (in $):
5n + 2(100 - n) > 300
5n + 200 - 2n > 300
3n > 100
n > 100/3
n > (33 and 1/3)

Since n is an integer, the least value of n = 34


(b)
If the number of $5 coins is doubled and the number of $2 coins is halved,
total amount of these coins
= $[5(2n) + 2(100 - n)/2]
= $(10n + 100 - n)
= $(9n + 100)

Since n ≥ 34,
total amount of these coins
= $(9n + 100)
≥ $(9*34 + 100)
= $406

Total amount of these coins ≥ 406
Hence, total amount of these coins will exceed $400.

(Since n is an integer, substitute n = 34 into the expression instead of n = (33 and 1/3).)

a) no. of $5 coins: n
no. of $2 coins: 100-n

so, 5n+2(100-n) >300
n>100/3(33.3)

so, the least value of n is 34.
(since n is a positive integer)


b) now, the least no. of $5 coins is 34*2=68

and the no. of $2 coins is (100-34)/2=33

the least total amount of these coins
=68*5+33*2
=$406
so, the total amount of these coins will exceed $400