Figure it Out
Q1. A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?
Answer:
Total parts = 3 + 4 + 3 + 5 = 15. Each part = 150 ÷ 15 = 10 minutes. Warm-up/cool-down = 3 × 10 = 30 min, batting = 4 × 10 = 40 min, bowling = 3 × 10 = 30 min, fielding = 5 × 10 = 50 min.
Q2. A school library has books in different languages in the following ratio — no. of Odiya books : no. of Hindi books : no. of English books :: 3 : 2 : 1. If the library has 288 Odiya books, how many Hindi and English books does it have?
Answer:
Odiya books = 3 parts = 288. So 1 part = 288 ÷ 3 = 96. Hindi books = 2 parts = 2 × 96 = 192. English books = 1 part = 96.
Q3. I have 100 coins in the ratio — no. of ₹10 coins : no. of ₹5 coins : no. of ₹2 coins : no. of ₹1 coins :: 4 : 3 : 2 : 1. How much money do I have in coins?
Answer:
Total parts = 4 + 3 + 2 + 1 = 10. Each part = 100 ÷ 10 = 10 coins. ₹10 coins = 4 × 10 = 40 coins → value = 40 × 10 = ₹400. ₹5 coins = 3 × 10 = 30 coins → value = 30 × 5 = ₹150. ₹2 coins = 2 × 10 = 20 coins → value = 20 × 2 = ₹40. ₹1 coins = 1 × 10 = 10 coins → value = 10 × 1 = ₹10. Total money = 400 + 150 + 40 + 10 = ₹600.
Q4. Construct a triangle with sidelengths in the ratio 3 : 4 : 5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?
Answer:
To construct: Draw a base of any length, say 3 cm. Using compass, draw arcs of 4 cm and 5 cm from the endpoints to form the triangle. All triangles with sides in ratio 3:4:5 are similar, not necessarily congruent, because the actual side lengths can be scaled (e.g., 3 cm, 4 cm, 5 cm or 6 cm, 8 cm, 10 cm). Congruence requires exact same side lengths, not just the same ratio.
Q5. Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?
Answer:
No, because for a triangle, the sum of any two sides must be greater than the third side. Here, 1 + 3 = 4, which is less than 5. So the triangle inequality is violated, and construction is impossible.
Q1. A group of 360 people were asked to vote for their favourite season from the three seasons — rainy, winter and summer. 90 liked the summer season, 120 liked the rainy season, and the rest liked the winter. Draw a pie chart to show this information.
Answer:
Total = 360. Summer = 90 → angle = (90/360) × 360° = 90°. Rainy = 120 → angle = (120/360) × 360° = 120°. Winter = 360 - (90+120) = 150 → angle = (150/360) × 360° = 150°. Draw a circle, mark radius, and draw sectors of 90°, 120°, and 150° successively. Label each sector with the season name.
Q2. Draw a pie chart based on the following information about viewers' favourite type of TV channel: Entertainment — 50%, Sports — 25%, News — 15%, Information — 10%.
Answer:
Entertainment: 50% of 360° = 180°. Sports: 25% of 360° = 90°. News: 15% of 360° = 54°. Information: 10% of 360° = 36°. Draw a circle, mark radius, and draw sectors of 180°, 90°, 54°, and 36° successively. Label each sector with the channel type.
Q3. Prepare a pie chart that shows the favourite subjects of the students in your class. You can collect the data of the number of students for each subject shown in the table (each student should choose only one subject). Then write these numbers in the table and construct a pie chart: Subject: Language, Arts Education, Vocational Education, Social Science, Physical Education, Maths, Science Number of Students: (blank to fill)
Answer:
Collect data from your class. For example, if total students = 40 and counts are: Language=8, Arts Ed=6, Voc Ed=4, Soc Sci=5, Phys Ed=7, Maths=5, Science=5. Total = 40. Calculate each angle: (count/40)×360°. Draw a circle and mark sectors accordingly. Label each sector with subject name. Since data varies, use actual class data.
Q1. Which of these are in inverse proportion? (i) x: 40, 80, 25, 16 y: 20, 10, 32, 50 (ii) x: 40, 80, 25, 16 y: 20, 10, 12.5, 8 (iii) x: 30, 90, 150, 10 y: 15, 5, 3, 45
Answer:
Two quantities are in inverse proportion if the product xy is constant for all pairs.
(i) Products: 40×20=800, 80×10=800, 25×32=800, 16×50=800. All equal. So yes, inverse proportion.
(ii) Products: 40×20=800, 80×10=800, 25×12.5=312.5, 16×8=128. Not constant. So no.
(iii) Products: 30×15=450, 90×5=450, 150×3=450, 10×45=450. All equal. So yes, inverse proportion.
Q2. Fill in the empty cells if x and y are in inverse proportion. x: 16, 12, 36 y: 9, ?, ?
Answer:
For inverse proportion, xy = constant. From first pair: 16×9=144. So constant is 144.
For x=12, y = 144/12 = 12.
For x=36, y = 144/36 = 4.
So the table: x: 16, 12, 36; y: 9, 12, 4.
Q1. Which of the following pairs of quantities are in inverse proportion? (i) The number of taps filling a water tank and the time taken to fill it. (ii) The number of painters hired and the days needed to paint a wall of fixed size. (iii) The distance a car can travel and the amount of petrol in the tank. (iv) The speed of a cyclist and the time taken to cover a fixed route. (v) The length of cloth bought and the price paid at a fixed rate per metre. (vi) The number of pages in a book and the time required to read it at a fixed reading speed.
Answer:
(i) Inverse proportion: More taps → less time to fill.
(ii) Inverse proportion: More painters → fewer days.
(iii) Direct proportion: More petrol → more distance (at constant mileage). Not inverse.
(iv) Inverse proportion: Higher speed → less time for fixed distance.
(v) Direct proportion: More cloth → more price at fixed rate. Not inverse.
(vi) Direct proportion: More pages → more time at fixed reading speed. Not inverse.
So inverse proportion pairs: (i), (ii), (iv).
Q2. If 24 pencils cost ₹120, how much will 20 such pencils cost?
Answer:
Cost per pencil = 120/24 = ₹5. So 20 pencils cost 20×5 = ₹100.
Q3. A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?
Answer:
Total families after move-in = 20+10 = 30. Water usage is inversely proportional to number of families (assuming same consumption per family per day). So 20×6 = 30×days → days = (20×6)/30 = 4 days.
Assumptions: Each family uses the same amount of water per day, and water is consumed at a constant rate.
Q4. Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.
Answer:
This question refers to a chart not provided in the text. Assuming typical averages: Human adults ~8 hours, human babies ~15 hours, cats ~13 hours, dogs ~10.5 hours, giraffes ~2.5 hours, etc. Without the chart, cannot give exact answer. Check the chart in the book for exact figures.
Q5. The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions. (i) What is the most common mode of transport? (ii) What fraction of children travel by car? (iii) If 18 children travel by car, how many children took part in the survey? How many children use taxis to travel to school? (iv) By which two modes of transport are equal numbers of children travelling?
Answer:
The pie chart shows angles: Walk 120°, Cycle 90°, Bus 60°, Two-wheeler 60°, Car 30° (since total 360°, and 120+90+60+60=330, so Car = 30°).
(i) Walk (largest angle 120°).
(ii) Car fraction = 30/360 = 1/12.
(iii) If 18 children = 1/12 of total, then total children = 18×12 = 216. Taxis are not mentioned in the chart; the chart shows only Walk, Cycle, Bus, Two-wheeler, Car. So 0 children use taxis.
(iv) Bus and Two-wheeler both have 60° angles, so equal numbers travel by bus and two-wheeler.
Q6. Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?
Answer:
Total work = 3 workers × 4 days = 12 worker-days. With 4 workers, days = 12/4 = 3 days.
Assumptions: All workers work at the same rate, and no other factors change.
Q7. It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?
Answer:
Time to fill one tank = 6/2 = 3 hours. So 5 tanks will take 5×3 = 15 hours.
Q8. A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?
Answer:
Total chairs = 25×12 = 300. With 20 chairs per row, number of rows = 300/20 = 15 rows.
Q9. A school has 8 periods a day, each of 45 minutes duration. How long is each period, if the school has 9 periods a day, assuming that the number of school hours per day stays the same?
Answer:
Total school hours = 8×45 = 360 minutes. With 9 periods, each period = 360/9 = 40 minutes.
Q10. A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?
Answer:
Small pump fills 1/3 tank per hour. Large pump fills 1/2 tank per hour. Combined rate = 1/3 + 1/2 = 5/6 tank per hour. Time to fill 1 tank = 1 / (5/6) = 6/5 hours = 1.2 hours (or 1 hour 12 minutes).
Q11. A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?
Answer:
Total machine-days = 42×63 = 2646. For 54 days, machines needed = 2646/54 = 49 machines.
Q12. A car takes 2 hours to reach a destination, travelling at a speed of 60 km/h. How long will the car take if it travels at a speed of 80 km/h?
Answer:
Distance = 60×2 = 120 km. Time at 80 km/h = 120/80 = 1.5 hours (or 1 hour 30 minutes).