CAT — Ratio & Proportion
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Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is
1/20
3/10
1/5
3/20
3/10
Here is the step-by-step breakdown to solve this problem without using LaTeX formatting:
1. Determine Efficiencies (Work Rates)
Let's find the ratio of work rates (efficiencies) for Sam, Mohit, and Ayna.
● Mohit is twice as fast as Sam, which means Sam's efficiency = Mohit's efficiency / 2.
● Mohit is thrice as fast as Ayna, which means Ayna's efficiency = Mohit's efficiency / 3.
To keep the calculations clean and avoid fractions early on, let's assume Mohit's efficiency is a common multiple of 2 and 3.
● Mohit's efficiency = 6 units/day
Using this assumption:
● Sam's efficiency = 6 / 2 = 3 units/day
● Ayna's efficiency = 6 / 3 = 2 units/day
2. Find Total Work
Sam can complete the entire job alone in 20 days.
● Total Work = Sam's efficiency * 20 days
● Total Work = 3 * 20 = 60 units
3. Analyze the 3-Day Work Cycle
The team works in a repeating 3-day pattern:
● Day 1 (Sam + Mohit): 3 + 6 = 9 units
● Day 2 (Sam + Ayna): 3 + 2 = 5 units
● Day 3 (Mohit + Ayna): 6 + 2 = 8 units
● Total work done in 1 full cycle (3 days): 9 + 5 + 8 = 22 units
4. Track Progress to Completion
Now, let's see how many full cycles fit into the total work of 60 units.
● After 2 full cycles (6 days): Work completed = 22 * 2 = 44 units
● Remaining work = 60 - 44 = 16 units
Now, we evaluate the subsequent days step-by-step:
● Day 7 (Sam + Mohit turn): They can complete 9 units.
● Remaining work = 16 - 9 = 7 units
● Day 8 (Sam + Ayna turn): They can complete 5 units.
● Remaining work = 7 - 5 = 2 units
● Day 9 (Mohit + Ayna turn): Only 2 units are left. Since their combined capacity is 8 units/day, they will finish this remaining work in 2/8 (or 1/4) of a day. Sam does not work on this day.
5. Calculate Sam's Contribution
Let's count the total number of days Sam actually worked:
● In the first 2 cycles (6 days): Sam works on Day 1 and Day 2 of each cycle.
● Days worked = 2 cycles * 2 days/cycle = 4 days
● Day 7: Sam works the full day. (+1 day)
● Day 8: Sam works the full day. (+1 day)
● Day 9: Sam does not work.
● Total days Sam worked: 4 + 1 + 1 = 6 days
Since Sam's efficiency is 3 units/day:
● Work done by Sam = 6 days * 3 units/day = 18 units
6. Find the Fraction of Total Work
● Fraction = Work done by Sam / Total Work
● Fraction = 18 / 60 = 3/10
Correct Answer:
B. 3/10
When Rajesh's age was same as the present age of Garima, the ratio of their ages was 3 : 2. When Garima's age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become
3 : 2
4 : 3
5 : 4
2 : 1
5 : 4
Given:
When Rajesh's age was the same as Garima's present age, the ratio of their ages was 3 : 2.
Find the ratio of their ages when Garima's age becomes equal to Rajesh's present age.
Step 1: Let the present ages be
Rajesh = R years
Garima = G years
where R > G.
Step 2: Use the first condition
When Rajesh's age was G years, he was
R − G
years younger than now.
At that time,
Garima's age = G − (R − G)
= 2G − R
Given,
G : (2G − R) = 3 : 2
Cross-multiplying,
2G = 3(2G − R)
2G = 6G − 3R
3R = 4G
Hence,
R : G = 4 : 3
Step 3: Let
R = 4x
and
G = 3x
Step 4: Find the required ratio
Garima reaches Rajesh's present age after
4x − 3x = x
years.
At that time,
Rajesh's age = 4x + x = 5x
Garima's age = 3x + x = 4x
Therefore, the required ratio is
5 : 4
Answer:
C. 5 : 4
In September, the incomes of Kamal, Amal and Vimal are in the ratio 8 : 6 : 5. They rent a house together, and Kamal pays 15%, Amal pays 12% and Vimal pays 18% of their respective incomes to cover the total house rent in that month. In October, the house rent remains unchanged while their incomes increase by 10%, 12% and 15%, respectively. In October, the percentage of their total income that will be paid as house rent, is nearest to
15.18
13.26
14.84
12.75
13.26
Step 1: Let incomes be 8x, 6x and 5x.
Find the house rent:
Kamal pays 15% of 8x = 1.2x
Amal pays 12% of 6x = 0.72x
Vimal pays 18% of 5x = 0.9x
Total house rent = 1.2x + 0.72x + 0.9x = 2.82x
Step 2: Find the total income in October
Kamal's income = 8x × 1.10 = 8.8x
Amal's income = 6x × 1.12 = 6.72x
Vimal's income = 5x × 1.15 = 5.75x
Total income = 8.8x + 6.72x + 5.75x = 21.27x
Step 3: Find the percentage of income spent on rent
= (2.82x / 21.27x) × 100 = (2.82 / 21.27) × 100 ≈ 13.26%
In a company, 20% of the employees work in the manufacturing department. If the total salary obtained by all the manufacturing employees is one-sixth of the total salary obtained by all the employees in the company, then the ratio of the average salary obtained by the manufacturing employees to the average salary obtained by the non-manufacturing employees is
6:5
4:5
5:4
5:6
4:5
Let total employees = 100.
Manufacturing = 20, Non-manufacturing = 80.
Step 1: Assume total salary
Let total salary = 6 units.
Manufacturing salary = 1 unit, Non-manufacturing salary = 5 units.
Step 2: Find average salaries
Average salary of manufacturing = 1/20
Average salary of non-manufacturing = 5/80 = 1/16
Step 3: Find the required ratio
= (1/20) : (1/16) = 16 : 20 = 4 : 5
The salaries of three friends Sita, Gita and Mita are initially in the ratio 5 : 6 : 7, respectively. In the first year, they get salary hikes of 20%, 25% and 20%, respectively. In the second year, Sita and Mita get salary hikes of 40% and 25%, respectively, and the salary of Gita becomes equal to the mean salary of the three friends. The salary hike of Gita in the second year is
25%
28%
26%
30%
26%
Let the initial salaries be
● Sita = 5x
● Gita = 6x
● Mita = 7x
After the first year
Sita's salary = 5x × 1.20 = 6x
Gita's salary = 6x × 1.25 = 7.5x
Mita's salary = 7x × 1.20 = 8.4x
After the second year
Sita gets a 40% hike. New salary = 6x × 1.40 = 8.4x
Mita gets a 25% hike. New salary = 8.4x × 1.25 = 10.5x
Let Gita's salary after the second year be G.
Given, Gita's salary becomes equal to the mean salary of all three friends.
So, G = (8.4x + G + 10.5x)/3
Multiply both sides by 3, 3G = 18.9x + G
2G = 18.9x
G = 9.45x
Find Gita's second-year salary hike
Salary before the second-year hike = 7.5x
Increase in salary = 9.45x − 7.5x = 1.95x
Therefore, Salary hike % = (1.95x ÷ 7.5x) × 100 = 26%
Answer:
C. 26%
Arvind travels from town A to town B, and Surbhi from town B to town A, both starting at the same time along the same route. After meeting each other, Arvind takes 6 hours to reach town B while Surbhi takes 24 hours to reach town A. If Arvind travelled at a speed of 54 km/h, then the distance, in km, between town A and town B is
Let Surbhi's speed be v km/h.
Arvind's speed = 54 km/h
After meeting,
● Arvind takes 6 hours to reach
B.
● Surbhi takes 24 hours to reach
A.
Hence, Distance from meeting point to B = 54 × 6 = 324 km
Distance from meeting point to A = 24v
For two people starting at the same time and meeting, (speed ratio) = (distance covered before meeting ratio)
Therefore, 54 : v = 24v : 324
Cross multiply, 54 × 324 = 24v² → v² = (54 × 324)/24 = 729 → v = 27 km/h
Distance from meeting point to A = 24 × 27 = 648 km
Distance from meeting point to B = 324 km
Therefore, total distance = 648 + 324 = 972 km
Answer: 972
Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
18
16
14
12
12
Step 1: Let Anil's speed = x km/h and Sunil's speed = y km/h.
They meet in 1.5 hours: (x + y) × 1.5 = 45 → x + y = 30
Step 2: Express the total travel times
Anil's total time = 45/x
Sunil's total time = 45/y
Given: 45/x = 45/y + 1.25
Step 3: Substitute y = 30 − x
45/x = 45/(30 − x) + 1.25
Multiply by 4: 180/x = 180/(30 − x) + 5
Multiply by x(30 − x):
180(30 − x) = 180x + 5x(30 − x)
5400 − 180x = 180x + 150x − 5x²
5400 = 510x − 5x²
5x² − 510x + 5400 = 0
x² − 102x + 1080 = 0
(x − 12)(x − 90) = 0
Since x + y = 30, x = 90 is not possible. Therefore x = 12 km/h
The ratio of expenditures of Lakshmi and Meenakshi is 2 : 3, and the ratio of income of Lakshmi to expenditure of Meenakshi is 6 : 7. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is 4 : 9, then the ratio of their incomes is
3:5
5:6
2:1
7:8
3:5
Step 1: Assume the expenditures. Let the expenditures of Lakshmi and Meenakshi be 2x and 3x, respectively.
Step 2: Find Lakshmi's income. Given, Income of Lakshmi : Expenditure of Meenakshi = 6 : 7. So, Lakshmi's income = (6/7) × 3x = 18x/7.
Step 3: Find Lakshmi's savings. Lakshmi's savings = Income − Expenditure = 18x/7 − 2x = 18x/7 − 14x/7 = 4x/7.
Step 4: Find Meenakshi's savings. Let Meenakshi's income be M. Then, Meenakshi's savings = M − 3x. Given, Lakshmi's savings : Meenakshi's savings = 4 : 9. So, (4x/7) : (M − 3x) = 4 : 9. Cancelling 4: x/7 : (M − 3x) = 1 : 9. Therefore, M − 3x = 9x/7 → M = 3x + 9x/7 = 21x/7 + 9x/7 = 30x/7.
Step 5: Find the required ratio. Lakshmi's income : Meenakshi's income = 18x/7 : 30x/7 = 18 : 30 = 3 : 5.
Rajesh and Vimal own 20 hectares and 30 hectares of agricultural land, respectively, which are entirely covered by wheat and mustard crops. The cultivation area of wheat and mustard in the land owned by Vimal are in the ratio of 5 : 3. If the total cultivation area of wheat and mustard are in the ratio 11 : 9, then the ratio of cultivation area of wheat and mustard in the land owned by Rajesh is
4 : 3
7 : 9
3 : 7
1 : 1
7 : 9
Given:
● Rajesh owns 20 hectares.
● Vimal owns 30 hectares.
● In Vimal's land, wheat : mustard = 5 : 3.
● Overall, wheat : mustard = 11 : 9.
Find the ratio of wheat to mustard in Rajesh's land.
Step 1: Find cultivation in Vimal's land
Total land = 30 hectares.
Since
Wheat : Mustard = 5 : 3,
Wheat
= (5/8) × 30
= 75/4 hectares
Mustard
= (3/8) × 30
= 45/4 hectares
Step 2: Let Rajesh's cultivation be
Wheat = x hectares
Mustard = 20 − x hectares
Step 3: Use the overall ratio
Total wheat
= x + 75/4
Total mustard
= (20 − x) + 45/4
Given,
(x + 75/4)/((20 − x) + 45/4) = 11/9
Step 4: Solve the equation
Simplify the denominator,
20 + 45/4
= 125/4
Hence,
(x + 75/4)/(125/4 − x) = 11/9
Cross-multiplying,
9(x + 75/4) = 11(125/4 − x)
9x + 675/4 = 1375/4 − 11x
20x = 700/4
20x = 175
x = 35/4
Step 5: Find the required ratio
Mustard area
= 20 − 35/4
= 45/4
Therefore,
Wheat : Mustard
= 35/4 : 45/4
= 7 : 9
Final Answer
Answer:
B. 7 : 9
An amount of Rs 10000 is deposited in bank A for a certain number of years at a simple interest of 5% per annum. On maturity, the total amount received is deposited in bank B for another 5 years at a simple interest of 6% per annum. If the interests received from bank A and bank B are in the ratio 10 : 13, then the investment period, in years, in bank A is
4
5
3
6
6
Step 1: Find the interest from Bank A
Interest from Bank A = (10000 × 5 × x)/100 = 500x
Maturity amount = 10000 + 500x
Step 2: Find the interest from Bank B
Interest from Bank B = [(10000 + 500x) × 6 × 5]/100 = 3000 + 150x
Step 3: Use the given ratio
500x : (3000 + 150x) = 10 : 13
13 × 500x = 10(3000 + 150x)
6500x = 30000 + 1500x
5000x = 30000 → x = 6
The ratio of the number of students in the morning shift and afternoon shift of a school was 13 : 9. After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 : 14. Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the number of students in the morning shift and the afternoon shift became 5 : 4. The number of new students who joined is
110
88
121
99
99
Let initial students be 13x and 9x. After shifting: (13x−21)/(9x+21) = 19/14 → 182x−294 = 171x+399 → x = 63. Morning = 798, Afternoon = 588.
Let new students join as 3k (morning) and 8k (afternoon). (798+3k)/(588+8k) = 5/4 → 3192+12k = 2940+40k → 28k = 252 → k = 9.
Total new students = 11k = 99.
− The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins that needs to be shifted further from A to B, to make this ratio 1:1, is
Step 1: Let initial coins in A = 17x, B = 7x.
After shifting 108: (17x−108)/(7x+108) = 37/20
Step 2: Cross-multiply
20(17x−108) = 37(7x+108)
340x − 2160 = 259x + 3996
81x = 6156 → x = 76
Step 3: After 108 coins shifted:
A = 17(76) − 108 = 1184, B = 7(76) + 108 = 640
Step 4: Let y more coins be shifted for 1:1 ratio.
1184 − y = 640 + y → 544 = 2y → y = 272
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