Revision and important questions: Proportional Reasoning-2

5.1 Formula box

1. The pair a : b and c : d is in proportion when a d = b c, which is the same as a/c = b/d.
2. For four-term ratios the matching quotients are all equal, so d/s = c/r = b/q = a/p.
3. Splitting a total x in the ratio a : b : c gives the shares x times a/(a + b + c), x times b/(a + b + c) and x times c/(a + b + c).
4. One part of a division is the whole divided by the sum of the ratio terms.
5. Map Representative Fraction: 1 : n means 1 unit on the map equals n of the same units on the ground. Convert centimetres to kilometres by dividing by 1,00,000.
6. Pie chart angle = (part/whole) × 360 degrees.
7. Direct proportion: both quantities change by the same factor, so y/x = k.
8. Inverse proportion: one quantity changes by a factor n and the other by 1/n, so x × y = k and x1 y1 = x2 y2.
9. Work rate: if one worker or pump finishes in t hours, the rate is 1/t of the job per hour. Add rates when they work together.

5.2 Common mistakes

1. Adding the terms wrongly. For the ratio 3 : 4 : 3 : 5 the sum is 15, not 12. Add every term, including repeated numbers.
2. Using the wrong total. A pie chart's total angle is always 360 degrees, not 100, even when the data is given in percentages.
3. Forgetting unit conversion on maps. An RF of 1 : 60,00,000 gives 60 km, not 60,00,000 km. Centimetres must be converted to kilometres.
4. Treating worker-days as a rate. 3 workers for 4 days is 12 worker-days in total, but the number of workers and the number of days are inversely proportional, not directly.
5. Adding times when people work together. Ram takes 1 hour and Shyam 1.5 hours, but together they take 3/5 hour. Rates are added, not times.
6. Ignoring the triangle inequality. Sides in the ratio 1 : 3 : 5 cannot form a triangle because 1 + 3 < 5.
7. Assuming congruence from a ratio. Sides in the ratio 3 : 4 : 5 give similar triangles of different sizes, not congruent ones.

5.3 Important exam-style questions

Question 1. Divide ₹840 in the ratio 3 : 4 : 7.

Answer. The sum of the terms is 3 + 4 + 7 = 14, so one part is 840/14 = ₹60. The shares are 3 × 60 = ₹180, 4 × 60 = ₹240 and 7 × 60 = ₹420. Check: 180 + 240 + 420 = 840.

Question 2. A map has scale 1 : 25,00,000. Two towns are 6 cm apart on the map. Find the geographical distance between them.

Answer. 6 cm on the map represents 6 × 25,00,000 = 1,50,00,000 cm. Divide by 1,00,000 to get kilometres: 1,50,00,000/1,00,000 = 150 km.

Question 3. In a school of 500 students, 60 choose cricket as their favourite sport. What angle will cricket have in a pie chart?

Answer. Angle = 60/500 × 360 = 0.12 × 360 = 43.2 degrees.

Question 4. Five machines produce a batch of toys in 12 days. How long will six machines take for the same batch?

Answer. The fixed work is 5 × 12 = 60 machine-days. With 6 machines, 6 × d = 60, so d = 10 days.

Question 5. Four taps fill a tank in 45 minutes. How long will nine taps take, if all flow at the same rate?

Answer. The fixed work is 4 × 45 = 180 tap-minutes. With 9 taps, 9 × t = 180, so t = 20 minutes.

Question 6. A 250 g mixture is made from three ingredients in the ratio 2 : 3 : 5. Find the weight of each ingredient.

Answer. Sum of terms = 10, so one part is 250/10 = 25 g. The weights are 2 × 25 = 50 g, 3 × 25 = 75 g and 5 × 25 = 125 g. Check: 50 + 75 + 125 = 250 g.

Question 7. Fifteen workers build a wall in 20 days. How many workers are needed to finish in 12 days?

Answer. The fixed work is 15 × 20 = 300 worker-days. For 12 days, workers = 300/12 = 25.

Question 8. State whether each pair is directly or inversely proportional: (a) the number of identical pipes and the time to empty a tank, (b) the number of identical books and their total weight, (c) a train's speed and the time for a route of fixed length, (d) the quantity of rice and its cost at a fixed rate.

Answer. (a) Inverse: more pipes empty the tank faster. (b) Direct: twice the books weigh twice as much. (c) Inverse: a faster train needs less time for the same route. (d) Direct: at a fixed rate, cost grows with quantity.

Question 9. Worker A finishes a task in 2 hours and worker B takes 3 hours. How long will they take together?

Answer. In one hour A does 1/2 and B does 1/3 of the task, so together they do 1/2 + 1/3 = 5/6. The time for the whole task is 1/(5/6) = 6/5 hours, which is 1 hour 12 minutes.

Question 10. In an inverse proportion table, x = 8 when y = 18. Complete the table for x = 12, 24 and for the value of x when y = 4.

Answer. The constant is 8 × 18 = 144. When x = 12, y = 144/12 = 12. When x = 24, y = 144/24 = 6. When y = 4, x = 144/4 = 36. The completed pairs are (8, 18), (12, 12), (24, 6) and (36, 4).