Olympiad Test: Comparing Quantities - Class 8 MCQ

Given:

• The amount at the end of the first year = Rs. 200
• The amount at the end of the second year = Rs. 220

We need to calculate the rate percent.

Step 1: Find the Compound Interest for the second year

The difference between the amount at the end of the second year (Rs. 220) and the amount at the end of the first year (Rs. 200) is the compound interest earned in the second year:

Compound Interest for 2nd year = 220 - 200 = Rs. 20

This Rs. 20 is the interest earned on Rs. 200 (the amount at the end of the first year).

Step 2: Use the formula for Compound Interest

Let the principal be P and the rate be r% per annum.

At the end of the first year, the amount is given by:

A₁ = P * (1 + r/100)

We are given that A₁ = 200, so:

P * (1 + r/100) = 200

At the end of the second year, the amount is given by:

A₂ = P * (1 + r/100)²

We are given that A₂ = 220, so:

P * (1 + r/100)² = 220

Step 3: Solve the equations

We already know that the compound interest for the second year is Rs. 20. The interest for the second year is the difference between the amount at the end of the second year and the amount at the end of the first year:

A₂ - A₁ = 220 - 200 = 20

Now, substitute A₁ = P * (1 + r/100) into the equation:

P * (1 + r/100) = 200

This gives:

P = 200 / (1 + r/100)

Now substitute P into A₂ = P * (1 + r/100)² = 220:

(200 / (1 + r/100)) * (1 + r/100)² = 220

Simplifying:

200 * (1 + r/100) = 220

Solving for r:

1 + r/100 = 220 / 200

1 + r/100 = 1.1

r/100 = 0.1

r = 10

Correct Option: The correct option is: C: 10%

Given:

• Sale price of the shirt = Rs. 176
• Discount = 20%

Let the marked price of the shirt be M.

Since a 20% discount is allowed, the sale price is 80% of the marked price. So, we can write:

Now, solve for M:

Thus, the marked price of the shirt is Rs. 220.

Correct option: D: Rs. 220

The problem involves finding the principal (sum of money) when the difference between Simple Interest (S.I.) and Compound Interest (C.I.) for 2 years at a rate of 15% p.a. is given as Rs. 144.

Formula for the difference between S.I. and C.I. for 2 years:

Where:

• P = Principal (sum of money to find)
• R = Rate of interest (15%=15/100=0.15)
• Difference = Rs. 144 (given)

Substituting values into the formula:

Final Answer:

The principal (sum of money) is Rs. 6400, which matches Option d).

Given:

• Principal (P) = Rs. 800
• Rate (R) = 5% per annum
• Amount (A) = Rs. 882

We can use the formula for compound interest:

A = P * (1 + R/100)^t

Where:

• A is the amount after t years,
• P is the principal,
• R is the rate of interest,
• t is the time in years.

Substitute the given values:

882 = 800 * (1 + 5/100)^t

Now simplify:

882 = 800 * (1.05)^t

Now divide both sides by 800:

882 / 800 = (1.05)^t

1.1025 = (1.05)^t

Take the logarithm of both sides:

log(1.1025) = t * log(1.05)

Using a calculator:

log(1.1025) ≈ 0.0436 and log(1.05) ≈ 0.0212

Now solve for t:

t = 0.0436 / 0.0212 ≈ 2.06

So, the time is approximately 2 years.

Correct Answer: D: 2 years

Given:

• Principal (P) = Rs. 8000
• Rate (R) = 5% per annum
• Time (T) = 1 year
• Interest is compounded half-yearly, so the rate for each half-year = 5%/2 = 2.5% per half year.
• Number of half-years (n) = 2

Formula for Compound Interest:

Where:

• A is the amount after time t,
• P is the principal,
• R is the annual interest rate,
• t is the time in years,
• n is the number of compounding periods per year.

Step-by-step Calculation:

1. Substitute the values into the formula:
   
2. Now calculate the Compound Interest:
   C.I. = A − P
   C.I. = 8405 − 8000 = 405

Therefore, the Compound Interest is Rs. 405.

Correct Answer: D: Rs. 405

Given:

• Principal (P) = Rs. 25,000
• Rate (R) = 20% per annum
• Time (T) = 1.5 years
• Compounded half-yearly, so the rate for each half-year = 20/2 = 10% per half year.
• Number of compounding periods (n) = 2 compounding periods per year, so for 1.5 years, the total number of periods (nt) = 3 periods.

Formula for Compound Interest:

Where:

• A is the amount after time t,
• P is the principal,
• R is the annual interest rate,
• t is the time in years,
• n is the number of compounding periods per year.

Step-by-step Calculation:

Substitute the values into the formula:

Therefore, the amount Raghu will need to pay after 1.5 years is Rs. 33,275.

Correct Answer: C: Rs. 33275​​​​​​​

Using formula for compound interest,

According to the question,

Amount = 1331, P = 1000

R = 10%

Substituting values in the above formula,

∴ It will take 3 years for principal amount of Rs. 1000 to become Rs. 1331 when compounded annually at 10% per annum.

Population after n years is calculated as:

P_n = P₀ × (1 + r₁) × (1 + r₂) × (1 + r₃)

Where:

• P₀ is the initial population,
• r₁, r₂, r₃ are the growth rates for each year.

Given:

• Initial population P₀ = 25000
• Growth rates:
  • Year 1: 4% = 0.04
  • Year 2: 5% = 0.05
  • Year 3: 8% = 0.08

Now, we can calculate the population after each year:

Year 1:

Population after Year 1 = 25000 × (1 + 0.04) = 25000 × 1.04 = 26000

Year 2:

Population after Year 2 = 26000 × (1 + 0.05) = 26000 × 1.05 = 27300

Year 3:

Population after Year 3 = 27300 × (1 + 0.08) = 27300 × 1.08 = 29484

Present Value = Original Value × (1 - R/100)^t

• Present Value = Rs. 9680
• Rate of depreciation (R) = 12% per annum
• Time (t) = 2 years

Step-by-Step Calculation:

9680 = Original Value × (1 - 12/100)² 9680 = Original Value × (0.88)² 9680 = Original Value × 0.7744

Original Value = 9680 / 0.7744 Original Value = 12500

1. Substitute the known values into the formula:
2. Solve for the Original Value:

The price at which it was purchased is Rs. 12,500.

Correct Answer: B: Rs. 12500

Value of the vehicle after 3 years:

Value = Rs. 175000 × (1 - 20/100)³
Value = Rs. 175000 × (0.80)³
Value = Rs. 175000 × 0.512
Value = Rs. 89600

Therefore, the total depreciation is:

Total Depreciation = Rs. (175000 - 89600) Total Depreciation = Rs. 85400

The total depreciation after 3 years is Rs. 85400.

Given that Simple interest in 2 years = Rs. 400

⇒ Simple interest for 1 year = Rs. 400/2 = Rs. 200

Hence Compound Interest for 1^styear = Rs 200

Givencompound interest for 2 years = Rs. 410

Therefore compound interest for 2 years = CI for 1^st year + CI for 2^nd year

⇒ Compound interest for 2^nd year = Rs. 410 – Rs. 200 = Rs. 210

that is interest on Rs. 200 for 1 year = Rs. 210 – Rs 200 = Rs. 10

⇒ 2R = 10

∴ R = 5%

Given:

Time = 2 years

Rate = 10%

C.I. = Rs. 525

Calculation:

C.I. = P[1 + (r/100)]^t - P

⇒ 525 = P[(11/10)² - 1]

⇒ 525 = P[21/100]

⇒ 2500 = P

New rate = 10% = 5%

Time = 4 years

S.I. = PRT/100

⇒ 2500 × 5 × 4/100

⇒ 25 × 20

⇒ 500

∴ S.I. will be Rs.500

Difference between CI and SI for 2 years = 696.30 – 660 = 36.30

SI for 2 years = 660

SI for 1 year = 660/2 = 330

∴ Interest rate = (36.3/330) × 100 = 11%

⇒ 16 = 2 × 2 × 2 × 2 = 2⁴

⇒ n = 4

⇒ Amount doubles in  3 years

⇒ x = 3

Time = n × x

⇒ Time = 4 × 3 =12 years

∴ The amount will become 16 times in 12 years.

Given:

Compound interest = Rs. 328, Rate = 5% and Time = 2 years

Formula used:

If P = Principal, R = Rate of Interest, T = Time period, then:

CI = P(1 + R/100)^T - P

SI = (P × R × T) /100

Calculation:

328 = P(1 + 5/100)² - P

⇒ 328 = (41/400)P

⇒ P = Rs. 3200

Now S.I = (3200 × 5 × 2)/100 = Rs. 320

∴ The simple interest of the sum is Rs. 320

The formula for compound interest is:

A = P * (1 + R/100)^t

Where:

• A = Amount after t years
• P = Principal amount
• R = Rate of interest
• t = Time in years

Step-by-step Solution:

Step 1: Calculate the amount after the first year.

For the first year, the interest rate is 15%. Using the compound interest formula:

A₁ = P * (1 + 15/100)
A₁ = 12500 * (1 + 0.15)
A₁ = 12500 * 1.15
A₁ = Rs. 14375

So, after the first year, the amount is Rs. 14375.

Step 2: Calculate the amount after the second year.

For the second year, the interest rate is 16%. The amount at the end of the first year becomes the principal for the second year.

A₂ = A1 * (1 + 16/100)
A₂ = 14375 * (1 + 0.16)
A₂ = 14375 * 1.16
A₂ = Rs. 16675

So, after the second year, the amount is Rs. 16675.

Answer: The amount after 2 years is Rs. 16675.

Answer: C: Rs. 16675

Given:

• Cost price (C.P.) = Rs. 16.20 per kg
• Selling price (S.P.) = Rs. 17.28 per kg

Formula for Gain Percent:

Where:

Gain = Selling Price − Cost Price

Step 1: Calculate the Gain

Gain = 17.28 − 16.20 = 1.08 Rs.

Step 2: Calculate the Gain Percent

Final Answer: The gain percent is 6.67% or .

Given:

• Cost of the first pack: 25 paise per toffee.
• Cost of the second pack: 3 toffees for 65 paise.
• Selling price: Rs. 3.50 a dozen (12 toffees).

Step 1: Calculate the cost price of each pack

Cost of the first pack:

Cost per toffee = 25 paise

Cost of the first pack = 25 × n paise (where n is the number of toffees in the pack)

Cost of the second pack:

Cost of 3 toffees = 65 paise

Cost of 1 toffee = 65 ÷ 3 = 21.67 paise

Cost of the second pack = (65 ÷ 3) × n = 21.67 × n paise

Step 2: Total cost price of the two packs

Total cost = 25n + (65n ÷ 3) = (75n ÷ 3) + (65n ÷ 3) = (140n ÷ 3) paise

Step 3: Selling price of the two packs

Selling price of 1 toffee = 350 ÷ 12 = 29.17 paise

Total selling price for 2n toffees = 29.17 × 2n = 58.34n paise

Step 4: Calculate the gain

Gain = Total selling price − Total cost price

Gain = 58.34n − (140n ÷ 3)

Gain = (175.02n ÷ 3) − (140n ÷ 3) = (35.02n ÷ 3) paise

Step 5: Calculate the gain percent

Gain percent = (Gain ÷ Total cost price) × 100

Gain percent = (35.02n ÷ 140n) × 100

Gain percent = (35.02 ÷ 140) × 100 = 25%

Final Answer: The gain percent is 25%.

Answer: B: 25

Given:

• Cost price of 542 kg of sugar = Rs. 7560.90
• Gain percentage = 20%

To find: Selling price per kilogram of sugar

Step 1: Find the Cost Price per kilogram

Cost price per kg = (Total cost price) / (Total quantity)

Cost price per kg = 7560.90 / 542 = 13.96 Rs.

Step 2: Find the Selling Price per kilogram

The man gains 20% on the cost price. So,

Selling price per kg = Cost price per kg + (20% of Cost price per kg)

Selling price per kg = 13.96 + (0.20 × 13.96)

Selling price per kg = 13.96 + 2.792 = 16.752 Rs.

Rounding to two decimal places:

Selling price per kg = 16.74 Rs.

Answer: A: Rs. 16.74

Let cost price = x
initial marked price = cost price + 30% of cost price
= X + 30%X = 130 % X
Final marked price = initial marked price - 25/4% of initial marked price
= 130%X - 25/4 % of 130 % X = 121.875 %
Therefore gain percentage = (121.875 - 100)% = 21.875%