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From the data:

Shop C: 64 umbrellas

Shop D: 80 umbrellas

Total umbrellas sold by C and D:

Total C + D = 64 + 80 = 144

From the data:

Shop B: 160 umbrellas

Now, let's calculate the percentage.

We use the formula:

Percentage = Part / Whole × 100

Here:

Part = Total umbrellas sold by C and D = 144

Whole = Umbrellas sold by B = 160

Substitute the values:

Percentage = 144 / 160× 100

Simplify:

Percentage = 0.9 × 100 = 90%

Thus, the total number of umbrellas sold by shops C and D together is 90% of the number of umbrellas sold by shop B.

Total umbrellas sold by Shop A.

From the data:

Total umbrellas sold by Shop A = 96

The ratio of folded to unfolded umbrellas is 5:3.

Let:

Folded umbrellas = 5x 

Unfolded umbrellas = 3x 

Total umbrellas:

5x + 3x = 8x

Given that the total umbrellas sold by Shop A is 96:

8x = 96

x = 12

Now:

Folded umbrellas = 5x = 5 × 12 = 60 

Unfolded umbrellas = 3x = 3 × 12 = 36 

3/4^th of folded umbrellas are black.

Black folded umbrellas:

3/4 × 60 = 45

2/3^th of unfolded umbrellas are black.

Black unfolded umbrellas:

2/3 × 36 = 24

Total black umbrellas sold by Shop A.

Total black umbrellas = Black folded umbrellas + Black unfolded umbrellas

Total black umbrellas = 45 + 24 = 69

Thus, the number of black umbrellas sold by Shop A is 69.

From the data, the total number of umbrellas sold by Shop D is 80.

It is given that the number of unfolded umbrellas sold by Shop D is 55% of the total umbrellas sold by Shop D. 

Unfolded umbrellas sold by D = 55% × 80 = 0.55 × 80 = 44

The total umbrellas sold by Shop D consist of folded and unfolded umbrellas. Therefore:

Folded umbrellas sold by D = Total umbrellas sold by D - Unfolded umbrellas sold by D

Folded umbrellas sold by D = 80 - 44 = 36

It is given that the total number of unsold folded umbrellas in Shop D is 5/9 of the total number of folded umbrellas sold by Shop D. 

Unsold folded umbrellas in D = 5 / 9 × Folded umbrellas sold by D

Unsold folded umbrellas in D = 5 / 9 × 36 = 20

The total number of unsold folded umbrellas in Shop D is 20.

From the data:

Shop A: 96 umbrellas

Shop C: 64 umbrellas

Shop E: 116 umbrellas

Total umbrellas A + C + E = 96 + 64 + 116

Total umbrellas A + C + E = 276

The average is calculated by dividing the total number of umbrellas by the number of shops 3 in this case:

Average = (Total umbrellas A + C + E) / Number of shops

Average = 276 / 3 = 92

Thus, the average of the total number of umbrellas sold by shops A, C, and E is 92.

Total umbrellas sold by Shop B = 160

Ratio of folded to unfolded umbrellas = 3:5

Let the number of folded umbrellas = 3x 

Let the number of unfolded umbrellas = 5x 

Total umbrellas = 3x + 5x = 8x 

So, 8x = 160 

Solving for x:

x = 160 / 8 = 20

Folded umbrellas sold by Shop B:

3x = 3 × 20 = 60

Total umbrellas sold by Shop D = 80

Ratio of folded to unfolded umbrellas = 1:3

Let the number of folded umbrellas = y 

Let the number of unfolded umbrellas = 3y 

Total umbrellas = y + 3y = 4y 

So, 4y = 80 

Solving for y:

y = 80 / 4 = 20

Folded umbrellas sold by Shop D:

y = 20

Folded umbrellas Shop B = 60

Folded umbrellas Shop D = 20

Total folded umbrellas:

60 + 20 = 80

Thus, the correct answer is 80.

Shortcut Trick

Sum of m and n = 36 × 2 = 72. So, n = 72 − m.

Ratio equation: (0.8m) / (72 − m + 13) = 9/10.

8m = 9 × (85 − m) ⇒ 8m = 765 − 9m ⇒ 17m = 765.

m = 765 ÷ 17 = 45.

∴ The correct answer is 45.

Alternate Method

Given:

Average of m and n = 36

Ratio [80% of m] : [n + 13] = 9 : 10

Calculations:

⇒ (m + n) / 2 = 36

⇒ m + n = 72

⇒ n = 72 − m

⇒ 80% of m = 0.8m = 4m/5

⇒ (4m/5) / (n + 13) = 9/10

⇒ (4m/5) / (72 − m + 13) = 9/10

⇒ (4m) / [5 × (85 − m)] = 9/10

⇒ 40m = 45 × (85 − m)

⇒ 8m = 9 × (85 − m)

⇒ 8m = 765 − 9m

⇒ 17m = 765

⇒ m = 45

∴ The correct answer is 45.

Additional Information

Average Concept

Average is the sum of observations divided by the number of observations. For two numbers, Average = (m + n) ÷ 2.

Percentage to Fraction Conversion

To simplify calculations, convert percentages to fractions: 80% = 80/100 = 4/5 or 0.8.

Ratio and Proportion

If a:b = c:d, then a/b = c/d. This allows for cross-multiplication to solve for unknown variables.

Shortcut Trick

Profit ratio is calculated as (Investment × Time).

Profit Ratio X : Y : Z = (4 × 3) : (6 × 4) : (5 × 6) = 12 : 24 : 30.

Simplified Profit Ratio = 2 : 4 : 5.

X's share (2 units) = Rs 18,000 ⇒ 1 unit = Rs 9,000.

Total profit (2 + 4 + 5 = 11 units) = 11 × 9,000 = Rs 99,000.

∴ The correct answer is Rs 99,000.

Alternate Method

Given → Investment ratio = 4 : 6 : 5 | Time ratio = 3 : 4 : 6 | X's profit = Rs 18,000. Formulas: Profit = Investment × Time | Total Profit = Sum of individual profit shares.

⇒ Calculate profit ratio for X, Y, and Z: (4 × 3) : (6 × 4) : (5 × 6) = 12 : 24 : 30. ⇒ Reducing the ratio by dividing by 6: 2 : 4 : 5. ⇒ Let the profit shares be 2k, 4k, and 5k. ⇒ Given X's share 2k = 18,000 ⇒ k = 9,000. ⇒ Total Profit = 2k + 4k + 5k = 11k. ⇒ Total Profit = 11 × 9,000 = 99,000.

∴ The correct answer is Rs 99,000.

Additional Information

Partnership Basics

In a partnership, the profit is distributed in the ratio of the product of Capital (Investment) and Time. If time is constant, profit is proportional to investment.

Ratio of Investments

If profits (P) and time (T) are known, the investment (I) ratio can be found using I = P ÷ T.

Shortcut Trick

Let the Cost Price (CP) be 100 units.

Marked Price (MP) = 100 + 50% of 100 = 150 units.

Selling Price (SP) = 150 − 20% of 150 = 150 − 30 = 120 units.

Profit = SP − CP = 120 − 100 = 20 units. Given 20 units = ₹660, so 1 unit = ₹33.

Discount = MP − SP = 150 − 120 = 30 units.

Difference between Discount and Profit = 30 − 20 = 10 units = 10 × 33 = ₹330.

∴ The correct answer is ₹330.

Alternate Method

Given: Markup% = 50%, Discount% = 20%, Profit = ₹660. Formulas: Profit = SP − CP | SP = MP × (100 − D%) ÷ 100 | MP = CP × (100 + Markup%) ÷ 100.

⇒ Let CP = x. Then MP = 1.5x and SP = 0.8 × 1.5x = 1.2x. ⇒ Profit = 1.2x − x = 0.2x. Given 0.2x = 660 ⇒ x = 3300. ⇒ Profit = 0.2 × 3300 = ₹660. ⇒ Discount = MP − SP = 1.5x − 1.2x = 0.3x = 0.3 × 3300 = ₹990. ⇒ Difference = 990 − 660 = ₹330.

∴ The correct answer is ₹330.

Additional Information

Markup and Profit Relationship

Net Profit% can be calculated using the successive percentage formula: Net% = m − d − (m × d)/100, where m is markup% and d is discount%.

Relation between CP and MP

The ratio of CP to MP is given by: CP/MP = (100 − Discount%)/(100 + Profit%).

Shortcut Trick

Ratio of Milk to Water = 6 : 1

Given Water = 15 litres (1 unit = 15 litres)

Total Milk (final) = 6 × 15 = 90 litres

Initial Milk (x) = Final Milk − Added Milk = 90 − 18 = 72 litres

∴ The correct answer is 72 litres.

Alternate Method

Given:

Initial Milk = x litres, Initial Water = 15 litres

Milk added = 18 litres

Final Ratio (Milk : Water) = 6 : 1

Calculations:

⇒ (Initial Milk + Added Milk) ÷ Water = 6 ÷ 1

⇒ (x + 18) ÷ 15 = 6 ÷ 1

⇒ x + 18 = 15 × 6

⇒ x + 18 = 90

⇒ x = 90 − 18

⇒ x = 72

∴ The correct answer is 72 litres.

Additional Information

Ratio and Proportion

A ratio compares two quantities. If a : b = c : d, then product of extremes (a×d) equals product of means (b×c).

Mixture Problems

When only one component is added to a mixture, the quantity of the other component remains constant in terms of absolute units.

Unitary Method

Find the value of one unit by dividing the given total by its ratio share, then multiply to find other values.

General Solution:

Let:

G_A : Questions given to A

G_B : Questions given to B

G_C : Questions given to C

A_A : Questions attempted by A

A_B : Questions attempted by B

A_C : Questions attempted by C

B attempted 50% of the questions he was given:

A_B = 0.5 × G_B

A was given 20 fewer questions than B:

G_A = G_B - 20

C was given 130 questions:

G_C = 130

Combined questions attempted by A and C is 145:

A_A + A_C = 145

Total questions given to all three is twice the total questions they attempted:

G_A + G_B + G_C = 2 × (A_A + A_B + A_C)

Number of questions attempted by B and C is the same:

A_B = A_C

From Equation 1:

A_B = 0.5 × G_B

From Equation 3:

G_C = 130

From Equation 2:

G_A = G_B - 20

From Equation 4:

A_A + A_C = 145

From Equation 5:

GA + G_B + G_C = 2 × (A_A + A_B + A_C)

Substitute G_A = G_B - 20 and G_C = 130 into Equation 5:

G_B - 20 + G_B + 130 = 2 × (A_A + A_B + A_C)

Simplify:

2G_B + 110 = 2 × (A_A + A_B + A_C)

G_B + 55 = A_A + A_B + A_C

From Equation 4:

A_A + A_C = 145

From Equation 6:

A_B = A_C

Substitute A_B = A_C into G_B + 55 = A_A + A_B + A_C:

G_B + 55 = AA + AB + AC

G_B + 55 = A_A + 2A_C

From Equation 4:

A_A = 145 - A_C

Substitute A_A = 145 - A_C into G_B + 55 = A_A + 2A_C:

G_B + 55 = 145 - A_C + 2A_C

Simplify:

G_B + 55 = 145 + A_C

GB = 90 + A_C

From Equation 1:

A_B = 0.5 × G_B

From Equation 6:

AB = A_C

So:

A_C = 0.5 × GB

Substitute A_C = 0.5 × G_B into G_B = 90 + A_C:

G_B = 90 + 0.5 × G_B

Subtract 0.5 × G_B from both sides:

0.5 × G_B = 90

G_B = 180

Now, find A_C:

A_C = 0.5 × G_B = 0.5 × 180 = 90

From Equation 6:

A_B = A_C = 90

From Equation 4:

A_A = 145 - AC = 145 - 90 = 55

From Equation 2:

G_A = G_B - 20 = 180 - 20 = 160

From Equation 3:

G_C = 130

Thus,

 

Calculations:

Questions attempted by A: 55

Questions given to B: 180

The ratio of A_A to G_B is:

A_A / G_B = 55 / 180

Divide both numerator and denominator by 5 to simplify:

55 / 180 = 11 / 36

Thus, the ratio of the number of questions attempted by A to the number of questions given to B is: 11:36