The original price of a car and the profit on it, twelve men and a job after four quit, a tree shadow thrown on a building, and distances between three moving cars

Understanding the topic

This question tests percentages, work rates, similar triangles and relative positions. Keep units visible and distinguish a remaining duration from total elapsed time. In the final part, convert 15 minutes to one-quarter of an hour before using the three speeds.

(a) A man purchases a car in an amount of Rs. 2400,000 in which he pays one-fourth extra as profit. Find the original amount of car and the amount of profit.

Interpret the printed amount Rs. 2400,000 as Rs. 2,400,000.

Let the original price be x. The profit is one-fourth of the original price:

Profit = x/4

The total price paid is:

x + x/4 = 2,400,000

5x/4 = 2,400,000

x = 2,400,000 × 4/5

x = Rs. 1,920,000

Profit:

Rs. 2,400,000 - Rs. 1,920,000 = Rs. 480,000

Check: one-fourth of Rs. 1,920,000 is Rs. 480,000, and their sum is Rs. 2,400,000.

Answer: Original price = Rs. 1,920,000; profit = Rs. 480,000.

(b) Twelve men can complete a job in twenty-four days. After four days four person quit. In how many days this job will be completed by the remaining persons.

Total work in man-days:

12 × 24 = 288 man-days

Work completed during the first four days:

12 × 4 = 48 man-days

Remaining work:

288 - 48 = 240 man-days

After four men leave, eight men remain. The additional time they require is:

240 ÷ 8 = 30 days

The phrase “by the remaining persons” asks for the time required by those eight workers after the four have left. That answer is 30 additional days. If total elapsed time from the beginning is required, add the first four days:

4 + 30 = 34 days

Answer: The remaining eight men complete the unfinished work in 30 more days, so the job finishes on day 34 from the original start.

(c) The shadow of a 10 m tall tree is falling on a high rise building and its height is 100 m. If the tree is 20 m away from the wall, at what distance from the wall is the light source?

Let x be the ground distance from the light source to the tree. The light source, tree and wall form two similar right triangles:

• Small triangle: height 10 m and base x
• Large triangle: height 100 m and base x + 20

By similarity:

10/x = 100/(x + 20)

Cross-multiply:

10(x + 20) = 100x

10x + 200 = 100x

90x = 200

x = 20/9 m ≈ 2.22 m

This is the distance from the light source to the tree. The question asks for the distance from the source to the wall:

x + 20 = 20/9 + 20

= 200/9 m ≈ 22.22 m

Answer: The light source is approximately 22.22 m from the wall.

(d) There are three cars and start moving in such a way that car A and B are moving opposite with speed 60 and 100 km/h. Car C is moving perpendicularly to both with speed 80 km/h. What is distance after 15 minutes between (i) A and B (ii) A and C (iii) B and C?

Convert the time:

15 minutes = 15/60 hour = 1/4 hour

Distances travelled in one-quarter hour:

• Car A: 60 × 1/4 = 15 km
• Car B: 100 × 1/4 = 25 km
• Car C: 80 × 1/4 = 20 km

(i) Distance between A and B

A and B move in opposite directions along the same line:

A-B distance = 15 + 25 = 40 km

(ii) Distance between A and C

Their paths are perpendicular:

A-C distance = √(15² + 20²)

= √625 = 25 km

(iii) Distance between B and C

Their paths are also perpendicular:

B-C distance = √(25² + 20²)

= √1025 = 5√41 km

≈ 32.02 km

Answers: A-B = 40 km; A-C = 25 km; B-C = 5√41 km, approximately 32.02 km.