Solution

Let’s assume the cost price (CP) of one item is Rs. 100.

Step 1: Calculate the selling price after markup and discount

– After a 96% markup, the price becomes Rs. 196.
– After a 25% discount, the selling price (SP) becomes \(196 \times \frac{3}{4} = Rs. 147\).

Step 2: Calculate the effective selling price for a dozen items with 2 items free

– When 2 items are given free on a dozen, the effective number of items sold is 10.
– Therefore, the effective selling price for a dozen items is \(10 \times 147 = Rs. 1470\).

Step 3: Calculate the cost price for a dozen items

– The cost price for a dozen items is \(12 \times 100 = Rs. 1200\).

Step 4: Calculate the profit and profit percent

– Profit = Selling price for a dozen – Cost price for a dozen = Rs. 1470 – Rs. 1200 = Rs. 270.
– Profit percent = \(\frac{\text{Profit}}{\text{Cost price for a dozen}} \times 100 = \frac{270}{1200} \times 100 = 22.5\%\).

Conclusion

The profit percent on the sale of 1 dozen items is 22.5%.

Solution

Given:
– The correct ratio for dividing the gold coins among Abhay, Vishal, and Kishore is 2:3:4.
– The mistaken ratio used for division is 1/2:1/3:1/4.
– There are a total of 351 gold coins.

Step 1: Calculate the number of coins each person should have received

Total parts in the correct ratio = 2 + 3 + 4 = 9 parts

– Abhay’s share in the correct ratio: \(\frac{2}{9} \times 351 = 78\) coins
– Vishal’s share in the correct ratio: \(\frac{3}{9} \times 351 = 117\) coins
– Kishore’s share in the correct ratio: \(\frac{4}{9} \times 351 = 156\) coins

Step 2: Calculate the number of coins each person received due to the mistake

Total parts in the mistaken ratio = 1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12 parts

– Abhay’s share in the mistaken ratio: \(\frac{1/2}{13/12} \times 351 = \frac{6}{13} \times 351 = 162\) coins
– Vishal’s share in the mistaken ratio: \(\frac{1/3}{13/12} \times 351 = \frac{4}{13} \times 351 = 108\) coins
– Kishore’s share in the mistaken ratio: \(\frac{1/4}{13/12} \times 351 = \frac{3}{13} \times 351 = 81\) coins

Step 3: Calculate the number of extra/deficit gold coins incurred to Abhay due to the mistake

Extra/deficit coins for Abhay = Abhay’s share in the mistaken ratio – Abhay’s share in the correct ratio
\[ = 162 – 78 = 84 \text{ coins} \]

Conclusion

The number of extra gold coins incurred to Abhay due to the mistake is 84 coins.

Solution

Given:
– The lengths of the sides of the triangle are 9 cm, 12 cm, and 15 cm.

We can use Heron’s formula to find the area of the triangle and then use the area to find the length of the perpendicular from the opposite vertex to the side of length 15 cm.

Step 1: Calculate the semi-perimeter (s) of the triangle

\[ s = \frac{9 + 12 + 15}{2} = 18 \text{ cm} \]

Step 2: Use Heron’s formula to find the area (A) of the triangle

\[ A = \sqrt{s(s – 9)(s – 12)(s – 15)} \]
\[ A = \sqrt{18(18 – 9)(18 – 12)(18 – 15)} \]
\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]
\[ A = \sqrt{2916} \]
\[ A = 54 \text{ cm}^2 \]

Step 3: Find the length of the perpendicular (h) from the opposite vertex to the side of length 15 cm

Using the formula for the area of a triangle (\(A = \frac{1}{2} \times \text{base} \times \text{height}\)):
\[ 54 = \frac{1}{2} \times 15 \times h \]
\[ h = \frac{54 \times 2}{15} \]
\[ h = \frac{108}{15} \]
\[ h = 7.2 \text{ cm} \]

Conclusion

The length of the perpendicular from the opposite vertex to the side whose length is 15 cm is 7.2 cm.

Solution

Given:
– The shop remains closed on Monday.
– The average sales per day from Tuesday to Sunday is Rs. 13,240.
– The average sales per day from Tuesday to Saturday is Rs. 13,924.

Step 1: Calculate the total sales from Tuesday to Sunday

Total sales from Tuesday to Sunday = Average sales per day (Tuesday to Sunday) × Number of days
\[ = 13240 \times 6 \]
\[ = Rs. 79,440 \]

Step 2: Calculate the total sales from Tuesday to Saturday

Total sales from Tuesday to Saturday = Average sales per day (Tuesday to Saturday) × Number of days
\[ = 13924 \times 5 \]
\[ = Rs. 69,620 \]

Step 3: Calculate the sales on Sunday

Sales on Sunday = Total sales from Tuesday to Sunday – Total sales from Tuesday to Saturday
\[ = 79,440 – 69,620 \]
\[ = Rs. 9,820 \]

Conclusion

The sales on Sunday are Rs 9,820.

Solution

Let’s denote the marks obtained by Anil, Barun, and Chandan as \(A\), \(B\), and \(C\) respectively.

Given:
– Anil’s marks are 28.57% less than Barun’s marks: \(A = B – 0.2857B = 0.7143B\).
– Barun’s marks are 11.11% less than Chandan’s marks: \(B = C – 0.1111C = 0.8889C\).
– The difference between the marks obtained by Anil and Chandan is 80.5: \(C – A = 80.5\).

Step 1: Express Anil’s marks in terms of Chandan’s marks

Using the relation between Anil’s and Barun’s marks:
\[ A = 0.7143B \]
Using the relation between Barun’s and Chandan’s marks:
\[ B = 0.8889C \]

So, Anil’s marks in terms of Chandan’s marks:
\[ A = 0.7143 \times 0.8889C = 0.6351C \]

Step 2: Use the difference between Anil’s and Chandan’s marks

\[ C – A = 80.5 \]
\[ C – 0.6351C = 80.5 \]
\[ 0.3649C = 80.5 \]
\[ C = \frac{80.5}{0.3649} \]
\[ C = 220.5 \]

Step 3: Find Barun’s marks

Using the relation between Barun’s and Chandan’s marks:
\[ B = 0.8889C \]
\[ B = 0.8889 \times 220.5 \]
\[ B = 196 \]

Conclusion

The marks obtained by Barun are 196.