In this chapter, we’ll cover important percentage and Ratio problems with quick tricks for time-bound competitive and entrance exams. We’ll discuss calculating percentages, percentage change, and finding the base value, along with useful tips to help you solve these problems with speed and accuracy. Let’s unlock the secrets to acing percentage problems in exams!
Assignment:1
Que 1: What is the range of values x can take if P is x% more than Q, Q is (x – 10)% less than R, and P is greater than R?
a) 10% to 28%
b) 10% to 25%
c) 10% to 37%
d) 10% to 43%
Solution: For 43%⇒R: Q: P= 100:67:95.xx
For 37%⇒R: Q: P= 100:73:100.01
Option (c) is correct
Que 2: What is the range of values x can take if an alloy of copper and aluminum has 40% copper, an alloy of copper and zinc has a copper-to-zinc ratio of 2:7, and when these two alloys are mixed in such a way that the overall alloy has more aluminum than zinc, with copper constituting x% of this alloy?
a) 30% ≤x ≤40%
b) 32.5% ≤x ≤42%
c) 33.33% ≤x ≤42%
d) 32.25% ≤x ≤40%
Solution- C: A=2:3
C: Z=2:7
Case 1: Aluminium and Zinc are equal
C: A=(2:3)×7=14:21⇒ 35 unit
C: Z=(2:7)×3=6:21⇒ 27 unit
Copper percentage= (20/62)×100= 32.25%
Case 2: The second alloy is negligible
C: A= 2:3
Copper percentage= 40%
Option (d) is correct
Que 3: What is the total amount earned by individuals A, B, C, D, and E, when A earns 25% more than B, C earns 25% more than A, A earns 20% more than D, and E earns 20% more than A, with all earnings being integers less than Rs. 100?
(a) 300 rs
(b) 245 rs
(c) 305 rs
(d) 480 rs
Solution: LCM(5,4,6)=60
| A | B | C | D | E | ||
| 5 | 4 | ×12 | ||||
| 4 | 5 | ×15 | ||||
| 6 | 5 | ×10 | ||||
| 5 | 6 | ×12 | ||||
| Final | 60 | 48 | 75 | 50 | 72 |
60+48+75+50+72=305 rs
Option (c) is correct
Que 4: If A, B, C, and D share a loot as follows: A receives a% of the total, B receives b% of the remaining (after A takes their share), C also receives c% of the remaining, and D gets the rest, where D’s share is a% less than what A gets, B and C get equal amounts, and b is equal to 2a. What percentage of what A received did C get?
(a) 160%
(b) 80%
(c) 175%
(d) 50%
Solution: Let the total amount= 100 rs
A⇒a
B⇒(100-a).b/100= (100-a).2a/100
C⇒(100-a).2a/100
D⇒a.(100-a)/100
A+B+C+D=100
5a.(120-a)=10000
a=20
A=20%=20
B=40% of 80=32
C=32
D= 80% of A=16
C received= 160% of the A
Option (a) is correct
Que 5: If A is x% less than B and A is y% less than C, and C is k% more than B, how can we express k in terms of x and y?
(a) \frac{(y-x)\times 100}{100-y}.
(b) \frac{(y+x)\times 100}{100-y}.
(c) \frac{(y-x)\times 100}{100+y}.
(a) \frac{(y+x)\times 100}{100+y}.
Solution: Let B=10, x=10%, and y=20%
A: B=(9:10)×4=36:40
A: C=(4:5)×9=36:45
A:B: C= 36:40:45
k=(45-40)×100/40= 12.5
Option (a) is correct
Que 6: What percentage of the student group are boys given that Class B has 50% more students than Class A, the number of girls in Class A is equal to the number of boys in Class B, and the percentage of girls is the same in both classes?
(a) 33.33%
(b) 40%
(c) 25%
(d) 60%
Solution: 100⇒x girls
150⇒x boys, 150-x girls
x% of 150= 100-x
x=60
Total boys=40+60=100
Percentage= 40%
Option (b) is correct
Que 7: What is the value of x if Trader A marks his goods up by x%, Trader B marks his goods up by 2x% and offers a discount of x%, and both traders make the same non-zero profit when buying goods for Rs. 1000 and Rs. 2000 respectively?
(a) 25%
(b) 12.5%
(c) 37.5%
(d) 40%
Solution: CP: SP=100:125
CP: SP=200⇒300⇒225
Option (a) is correct
Que 8: What is the minimum profit the merchant could have made on the trade if he buys 80 articles, each at Rs. 40, and sells n of them at a profit of n% and the remaining at a profit of (100 – n)%?
(a) 2160 rs
(b) 1420 rs
(c) 1580 rs
(d) 2210 rs
Solution: n% of 40= 2n/5
(100-n)% of 40=2.(100-n)/5
Profit= 2/5.[n2+(80-n).(100-n)]= 4/5.[n2-90n+4000]
n2-90n+4000= (n-45)2+1975
For minimum value: n=45
Put n=45
Profit=2/5.[2025+35×55]=1580 rs
Option (c) is correct
Que 9: What is the relation between x and y if A is x% more than B, B is y% less than C, and A, B, and C are positive with A being greater than C?
(a) y.(100-x)<100x
(b) y.(100-x)>100x
(c) y.(100+x)<100x
(d) y.(100+x)>100x
Solution: C=100
B=100-y
A=(100-y). (100+x)/100
A>C
10000-100y+100x-xy>10000
y.(100+x)<100x
Option (c) is correct
Que 10: What is the value of x if a two-digit number ab is 60% of x, and the two-digit number formed by reversing the digits of ab is 60% more than x?
(a) 45
(b) 54
(c) 63
(d) 72
Solution: 10a+b=3x/5
10b+a=8x/5
After division⇒ a:b=2:7
3x/5=27
x=45
Option (a) is correct
Assignment:2
Que 1: Find the tax percentage, if a 19% increase in tax results in a 1% decrease in the income of a person.
(a) 5%
(b) 2%
(c) 12.5%
(d) 20%
Solution: Use options
Let, tax= 5% and the total Salary is 100 rs.
Income= 100-5=95
Now 19% of 5= 1% of 95
Option (a) is correct.
Traditional method: 19% of r= 1% of (100-r)
0.19r=1-0.01r
0.2r=1
r=5
Que 2: In how many subjects did the student pass the examination if they obtained an aggregate of 60% marks in five subjects, given that the marks are in the ratio 10: 9: 8: 7: 6 and the passing marks are 50% of the maximum marks for each subject?
(a) 2
(b) 3
(c) 4
(d) 5
Solution: Average of (10,9,8,7,6)=8
60%=8
50%=6.67
So, passed 4 subjects
Option (c) is correct.
Que 3: What is the maximum number of bogies that can be attached to the train so that it can move, considering that the reduction in the speed of an engine is directly proportional to the square of the number of bogies attached to it? The speed of the train is 100 km/hr when there are 4 bogies and 55 km/hr when there are 5 bogies.
(a) 6
(b) 5
(c) 4
(d) None
Solution: Let the initial speed (without bogies)=s
Reduction in Speed (R)∝b2
R=kb2
Here k is a constant and b is the number of bogies
100=s-R=s-k.42
55=s-k.52
Subtract equation (ii) from (i):
45=9k
k=5
s=180 km/h
For zero speed⇒ s=R=k.b2
180=5.b2
b=6
The maximum number of bogies that can be attached=5
Option (b) is correct.
Que 4: What is the minimum number of passengers required for Arvind Singh to avoid any loss in his bus service, considering that his profit (P) depends on the number of passengers exceeding a certain minimum number (n) and the distance traveled by bus? Arvind’s profit is Rs. 3600 with 29 passengers on a journey of 36 km and Rs. 6300 with 36 passengers on a journey of 42 km.
(a) 12
(b) 20
(c) 18
(d) 15
Solution: Let d be the distance and m be the number of passengers. K is the proportionality constant.
P∝(m-n).d
P=k.(m-n).d
3600=k.(29-n).36
6300=k.(36-n).42
Divide equation (ii) by (i):
n=15
Option (d) is correct.
Que 5: What is the nearest percentage by which the weight of chocolate in the large box exceeds the weight of chocolate in the small box, given that identical chocolate pieces are sold in boxes of two sizes (small and large), with the large box being sold for twice the price of the small box? If the selling price per gram of chocolate in the large box is 12% less than that in the small box.
(a) 127
(b) 135
(c) 144
(d) 124
Solution: 12%=12/100=3/25
Let the price of each chocolate in the small box be 25 rs and in the large box be 22 rs
22x: 25y=2:1
x:y=25:11
Required percentage= [(25-11)/11]×100≈127%
Option (a) is correct.
Que 6: If a team has played 40 matches in a tournament and won 30% of them, if they win the remaining 60% of matches, then their overall winning percentage becomes 50%. If the team wins 90% of the remaining matches, then what will be the total number of matches won by the team?
(a) 78
(b) 84
(c) 80
(d) 86
Solution: Let the rest number of matches be x
30% of 40=12
(12+60% of x) :(40+x)=1: 2
x=80
The total number of matches won by the team=12+ 90% of 80=84
Option (b) is correct.
Que 7: If Shepard bought nine dozen goats at the end of 1998 and added p% of the goats at the beginning of each year while selling q% of the goats at the end of each year, which of the following statements is true if Shepard had nine dozen goats remaining at the end of year 2002, after making the sales for that year?
(a) p=q
(b) p<q
(c) p>q
(d) p=q/2
Solution: Shepard bought 4 times and sold 4 times
p<q, not possible
p=q not possible
p>q, correct
Option (c) is correct.
Que 8: What percentage of marks did Geeta score in the final examination, if Vishnu scored 52%, Asha scored 64%, and Geeta scored 84%? Additionally, it is known that the marks obtained by Vishnu are 23 less and the marks obtained by Asha are 34 more than the marks obtained by Ramesh.
(a) 399
(b) 439
(c) 417
(d) 357
Solution: 64%-34=52%+23
12%=57
84%=57×7=399
Option (a) is correct.
Que 9: If Amala’s income is 20% more than Bimala’s income and 20% less than Kamala’s income, what would be the nearest percentage by which Kamala’s income would exceed Bimala’s income if Kamala’s income goes down by 4% and Bimala’s income goes up by 10%?
(a) 31
(b) 29
(c) 28
(d) 32
Solution: A: B: K=12: 10: 15
B⇒11
K⇒14.4
Required percentage= 340/11≈31%
Option (a) is correct.
Que 10: If the salaries of Ramesh, Ganesh, and Rajesh were in the ratio 6: 5: 7 in 2010 and in the ratio 3: 4: 3 in 2015, and Ramesh’s salary increased by 25% during the period from 2010 to 2015, what would be the closest percentage increase in Rajesh’s salary during the same period?
(a) 7
(b) 8
(c) 9
(d) 10
Solution- R: G: Ra=60: 50: 70
Ramesh’s salary: 60⇒75
New ratio= 75: 100: 75
Increment in Rajesh’s Salary= 500/70≈7%
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