Explanation

- Brightest 20% = 20 students, mean score = 80  

- Dullest 25% = 25 students, mean score = 31  

- Remaining 55% = 55 students, mean score = x (to find)  

- Overall mean = 52

Using weighted average:

(20 × 80) + (25 × 31) + (55 × x) = 100 × 52  

1600 + 775 + 55x = 5200  

55x = 5200 - 2375  

55x = 2825  

x = 2825 / 55  

x = 51.36 approx.

Explanation

Let the original number of boys be n.

Original average = 30 kg  

New average after adding boy = 31 kg  

Weight of new boy = 35 kg

Total weight originally = 30n  

Total weight after adding = 30n + 35

New average = (30n + 35) / (n + 1) = 31

So,  

(30n + 35) = 31(n + 1)  

30n + 35 = 31n + 31  

35 - 31 = 31n - 30n  

4 = n

Explanation

Given: Time downstream is half the time upstream.

Let:

- Speed of boat in still water = b

- Speed of current = c

Speed downstream = b + c  

Speed upstream = b - c

Time ∝ 1 / Speed

Given: Time downstream = (1/2) × Time upstream  

=> 1/(b + c) = (1/2) × 1/(b - c)  

=> 2/(b + c) = 1/(b - c)  

=> 2(b - c) = b + c  

=> 2b - 2c = b + c  

=> 2b - b = c + 2c  

=> b = 3c  

=> b : c = 3 : 1

Explanation

Initially, the total weight of four boys is:

Total weight = Average weight * Number of boys

= 24 * 4

= 96 kg

When the fifth boy's weight is included, the new total weight is:

New total weight = New average weight * New number of boys

= 24.6 * 5

= 123 kg

The weight of the fifth boy is:

Weight of fifth boy = New total weight - Initial total weight

= 123 - 96

= 27 kg

Explanation

If price increases by 20%, new price = 120% of old price.  

To keep expenditure same, consumption must decrease to 100/120 = 5/6 = 83.33% of original.  

Reduction = 100% - 83.33% = 16.66% 

Explanation

Given a 12.5% reduction, the new price is 87.5% of the original price.

So,  

87.5% of x = 4375  

(87.5 / 100) × x = 4375  

0.875 × x = 4375  

Now, solve for x:  

x = 4375 / 0.875  

x = 5000

Explanation

Let's solve step-by-step:

- A’s rate = 1/15 work/day  

- B’s rate = 1/10 work/day  

Work done in the first 2 days (both A and B):  

(A + B) × 2 = (1/15 + 1/10) × 2 = (2/30 + 3/30) × 2 = (5/30) × 2 = (1/6) × 2 = 1/3 of the work.

Remaining work after 2 days:  

1 - 1/3 = 2/3

Time taken by A alone to complete the remaining work:  

Work = Rate × Time  

Time = Work / Rate = (2/3) / (1/15) = (2/3) × 15 = 10 days

Total time to complete the work:  

2 days (both working) + 10 days (A alone) = 12 days

Explanation

Use the formula:

% decrease = (increase ÷ (100 + increase)) × 100

= (25 ÷ 125) × 100 = 20%

So, to keep the total expenditure the same, consumption must decrease by 20%.

Explanation

Given:

- 6 men make 10 sofas in 2 days.

First, find the total man-days needed for 10 sofas:

6 men × 2 days = 12 man-days for 10 sofas.

So, man-days per sofa = 12 ÷ 10 = 1.2 man-days per sofa.

Now, for 8 sofas:

Total man-days needed = 8 sofas × 1.2 = 9.6 man-days.

If 8 men are working, number of days needed = 9.6 ÷ 8 = 1.2 days

Explanation

To find the original price before the 20% increase, follow these steps:

1. Let the original price be  P. The price is increased by 20%, so the new price is 120% of the original price.

New Price=1.20×Original Price

2. Set up the equation using the given new price of $60:

1.20×P=60

3. Solve for 

 𝑃= 60/ 1.20 = 50

So, the original price was $50.