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Class A (40 students) averages 60; class B (60 students) averages 70. Find the combined average.
(60 × 40 + 70 × 60) ÷ 100 = (2400 + 4200) ÷ 100 = 66.
Answer: 66
Three weeks of 5 working days have averages 40, 50, 60 minutes. Find the overall average.
Equal sizes: (40 + 50 + 60) ÷ 3 = 50 minutes.
Answer: 50 min
Mean of 9 numbers is 20. A tenth number 30 is added. New mean?
(180 + 30) ÷ 10 = 21.
Answer: 21
Mean of 8 numbers is 25. One number 41 is removed. New mean?
Sum = 200; 200 − 41 = 159; 159 ÷ 7 ≈ 22.71.
Answer: ≈ 22.71
Averages: 12 girls average 150 cm, group of 8 boys average h cm; whole group of 20 averages 154 cm. Find h.
12 × 150 + 8h = 20 × 154: 1800 + 8h = 3080, h = 160.
Answer: 160 cm
The combined average of 20 men (avg 55 kg) and 30 women (avg 50 kg): without calculating, is it above 52.5? Then compute.
More women, so it is closer to 50: below 52.5.
(1100 + 1500) ÷ 50 = 52, which is below 52.5.
Answer: 52 kg
200 mL of 30% syrup is mixed with 300 mL of 10% syrup. Concentration?
(60 + 30) ÷ 500 = 0.18.
Answer: 18%
How much water must be added to 4 litres of 20% juice to make it 16%?
Juice = 0.8 L; 0.8 ÷ (4 + x) = 0.16, so 4 + x = 5, x = 1 L.
Answer: 1 litre
Gold 18 g at 75% pure and 12 g at 90% pure are melted. Purity of the mix?
(13.5 + 10.8) ÷ 30 = 24.3 ÷ 30 = 0.81.
Answer: 81%
Can mixing 10% and 20% solutions give 25%?
No. A mixture always lies between the smallest and largest concentrations (10%–20%).
Answer: No
How many litres of 50% solution must be mixed with 6 litres of 20% to get 30%?
0.5x + 1.2 = 0.3(x + 6): 0.2x = 0.6, x = 3.
Answer: 3 litres
Marks: test 80%, project 70%, final exam 90% with weights 2 : 3 : 5. Annual score?
(160 + 210 + 450) ÷ 10 = 82%.
Answer: 82%
Marks: 40 out of 50 (weight 1), 45 out of 60 (weight 2), 78 out of 100 (weight 3). Weighted percentage?
80%, 75%, 78%: (80 + 150 + 234) ÷ 6 = 77.33%.
Answer: 77.33%
A shop records sales of three products over three months. To compare monthly totals and the contribution of each product, which chart?
A stacked bar chart (totals matter). Use the 100% chart if only the shares are of interest.
Answer: stacked bar
A family spends ₹6000 on food, ₹3000 on rent, ₹1000 on others. Give the 100% stacked bar shares.
Total 10000: 60%, 30%, 10%.
Answer: 60, 30, 10
Marks 60 and 80 with weights 1 and 3. What happens if the weights become 2 and 6?
Before: (60 + 240) ÷ 4 = 75. After: (120 + 480) ÷ 8 = 75. Unchanged.
Answer: 75, unchanged
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Method 1 gives each group equal importance, but there are 8 seniors and only 3 juniors. The senior average must count 8 times, the junior average 3 times. Correct: (165.5 × 8 + 149.33 × 3) ÷ 11 = 1772 ÷ 11 = 161.09 cm.
Weeks have 5 days each: (12.8 + 15.8 + 18) ÷ 3 = 15.53 min. It works when all collections are the same size, since (ap + bp + cp) ÷ 3p = (a + b + c) ÷ 3. It fails if the sizes differ.
(0.1y + 0.2y) ÷ 2y = 0.15 = 15%, midway.
(500 × 0.1 + 200 × 0.2) ÷ 700 = 90 ÷ 700 ≈ 13%, closer to 10% because the larger quantity has 10%.
0.55(320 + y) = 140 + 60 + 0.45y ⇒ y = 240 kg.
(60 × 3 + 64 × 2 + 73 × 5) ÷ 10 = 673 ÷ 10 = 67.3%.
1. Family C (950). 2. Family A (1770). 3. About ₹1280. 4. Housing (₹1750). 5. Healthcare (1770 > 1700). 6. Family C (7040; A = 8425, B = 7286). 7. About ₹7040. 8. 950 ÷ 7040 ≈ 13.5%. Questions 1–2 are easier on Choice 1; 4–5 on Choice 2; 6–8 on the stacked bar.
2005 (total 1000): 36%, 30%, 12%, 22%. 2025 (total 2000): 220 ⇒ 11%, 800 ⇒ 40%, 460 ⇒ 23%, 520 ⇒ 26%. Cooling’s share grew from 30% to 40% while lighting’s fell, and the total doubled.
Only (i). (ii) and (iii) compare numbers across gardens, but only percentages are shown (Case 1 and Case 2 show both outcomes). (iv) cannot be said because each bar is 100% whatever the total.
1. 100% bars show shares only; stacked bars show amounts and totals. 2. Yes: divide each part by its total. 3. No: the totals are lost. 4. Stacked: compare totals and amounts; 100%: compare shares within a bar.
The strip is for the average Indian, so it hides differences. Age groups show: children spend the most on learning, youth less (work and leisure grow), adults spend more on unpaid work and paid work, the elderly sleep and rest more. Sleep shown by age fits the trend studied last year. A strip with tiny segments clubbed by activity gives a 100% stacked bar.
(72 × 30 + 76 × 25) ÷ 55 = 4060 ÷ 55 ≈ 73.82.
(0.1 + 0.18 + 0.05) ÷ 3 = 0.33 ÷ 3 = 0.11 = 11/100.
(108 + 50 + 187) ÷ 31 = 345 ÷ 31 ≈ 11.13 varṇa.
Weights are the days: (3.5 × 31 + 10 × 30 + 8.7 × 31) ÷ 92 ≈ 7.37 mm. (A plain average of the three is only an approximation, as the months differ in length.)
(i) (100 × 5 + 200 × 10 + 300 × 15) ÷ 600 = 7000 ÷ 600 ≈ 11.67%. (ii) (300 × 5 + 200 × 10 + 100 × 15) ÷ 600 = 5000 ÷ 600 ≈ 8.33%.
Convert each mark to a percentage first. (iii) and (iv) are correct. (i) uses raw marks and (ii) divides marks out of different totals by 100. Value ≈ (210 + 293.33 + 400) ÷ 12 ≈ 75.28%.
(2400 + 500) ÷ 9000 ≈ 32.2%.
(44.5 × 20 + 55) ÷ 21 = 945 ÷ 21 = 45 km.
Salt: 30 mL in 900 mL ≈ 3.33%. Sugar: 24 mL in 900 mL ≈ 2.67%. Answer (v).
Spice 0.8 L: 0.8 ÷ (10 + x) = 0.06 ⇒ x = 10/3 ≈ 3.33 litres of water.
(i) Keerthi: she loses 5 in strength (weight 4) = −20 but gains 5 in agility (weight 6) = +30, net +10. (ii) Rashi (240 + 325 + 420) ÷ 15 ≈ 65.67; Keerthi (220 + 325 + 450) ÷ 15 ≈ 66.33.
Food (25 + 12 + 6) ÷ 10 = 4.3; ambience (16 + 15 + 2) ÷ 10 = 3.3; service (5 + 8 + 6 + 8 + 1) ÷ 10 = 2.8. Weighted: (6 × 4.3 + 5 × 3.3 + 4 × 2.8) ÷ 15 = 53.5 ÷ 15 ≈ 3.57.
Without computing: 14.925 is nearer to 13.8 than to 16.5, so there are more females. (i) (a). (ii) 16.5x + 13.8(60 − x) = 895.5 ⇒ 2.7x = 67.5 ⇒ 25 males (35 females). (iii) (13.8 × 35 + 15.2) ÷ 36 ≈ 13.84 kg. (iv) (412.5 − 33) ÷ 23 = 16.5 kg. (v) (412.5 − 1) ÷ 25 = 16.46 kg.
(i) 34 ÷ 3 ≈ 11.33%. (ii) Yes: 0.01% lies between 0.001% and 34%. 34 + 0.001x = 0.01(1 + x) ⇒ x ≈ 3777 litres of purified water. (iii) No: 0.001% is below both 34% and 0.01%, and a mixture lies between the two.
(i) The total number of animal species in the list in 2019. (ii) About 1000 reptile species in 2016 (read from the segment). (iii) Classes with the smallest segments growing, such as mammals and amphibians (approximate reading).
(i) Draw two bars of length 194 (home = 62 + 100 + 32) and 292 (overseas = 172 + 49 + 71). (ii) Three bars: Test 234 (62 home + 172 overseas), ODI 149 (100 + 49), T20 103 (32 + 71).
(i) A reasonable guess for a student: 8, 7, 9. (ii) Julie (8, 10, 6) sits a lot: a desk job (e.g. accountant, software engineer). (iii) 100% bars: Pavani 83%, 12.5%, 4%; Raghu 29%, 17%, 54%; Zakir 33%, 25%, 42%; Sahana ≈ 33%, 29%, 38%; Julie 33%, 42%, 25%.
1. Similar but not identical. 2. Yes: weekends include more lying down or moving around. 3. e.g. a bus driver may also sit much and a receptionist also looks like Julie.
(i) (6 + 12) ÷ 2 = 9. (ii) 19 overs at rate 6 = 114 runs; plus 12 gives 126 ÷ 20 = 6.3.
They hold 6, 4, 5, 4, 4 prices (23 in all). (i) Combined average = (6aᵧ + 4aₛ + 5aₖ + 4aₚ + 4a_g) ÷ 23. (ii) 13 nylon and 10 feather prices: (13aₙ + 10a_f) ÷ 23. They are equal (both are total ÷ 23). Here aₙ ≈ 109.92, a_f = 263.4, overall ≈ ₹176.65.
(i) (3750 + 300) ÷ 35 ≈ ₹115.71. (ii) (750 + 30n) ÷ (5 + n) = 70 ⇒ n = 10 shares.
(36 + 35 + 42 + 21 + 16) ÷ 30 = 150 ÷ 30 = 5 hastas.
(i) 22.5k. (ii) 25k. (iii) 12.5k. (iv) ≈ 10.45k. (v) 20k. (vi) 15k. (Each new average lies between 15k and the new price, nearer to the price when more grams are bought.)
The weighted mean is unchanged: (2w₁x₁ + … + 2wₙxₙ) ÷ (2w₁ + … + 2wₙ) = 2(Σwx) ÷ 2(Σw).
(i) Roughly 13,000 objects in 2003 and about twice that around 2021 (approximate readings). (ii) In 2025 both payload and other objects are about 17,000, about 50% each. (iii) Both grow over time; the number of other objects (debris, rocket stages) rose faster after 2007, so the share of payload objects first dominates and then falls.
(i) Only (b) (about 2 in 3). (a), (c), (d) mix percentages and counts: they need the number of schools. (ii) No. We need the number of schools in each state (to weight the percentages).
(i) On shares, Play A has the largest share of 4–5 star ratings. (ii) The stacked chart shows A has few ratings in total and B many, so decisions based on A’s share are less reliable; B’s large count of good ratings may change the decision. Shares hide sample size.
(i) A stacked bar chart (times add up to the total race time). (ii) Only (b) can be read from the 100% chart (A1: 300 ÷ 563 ≈ 53% cycling). Totals: A1 9:23, A2 9:50, A3 11:07, so (a) needs totals and (c) needs actual durations.
Shares only: movement-related disability forms a larger share in 60+; “any other” and “multiple” are bigger shares among 0–19. It does not tell how many people are in each group. We may wonder about causes and about actual numbers.
12: record hours, draw stacked bar(s) or 100% bars of expenditure categories, then write observations. 13: weights set by importance; compute each person’s rating Σwx ÷ Σw. 14: class average age = Σ(family average × members) ÷ Σ members.
New mean = (Σwx + Σx) ÷ (Σw + n). It generally changes, moving towards the simple mean. Example: values 10, 20 with weights 1, 3 give 17.5; with weights 2, 4 give 16.67. It stays the same only if the weighted mean equals the simple mean.
Shares must always add to 100%, so all three cannot decrease (i) or all increase (ii). Possible: (iii), (iv), (v).