Advanced Arithmetic & Algebra — Worked Practice
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5A.1 Quadratic Equation Comparison — A Worked Example
Given two equations: x² − 7x + 12 = 0 and y² − 9y + 20 = 0. Factorising the first: x² − 7x + 12 = (x−3)(x−4) = 0, so x = 3 or x = 4. Factorising the second: y² − 9y + 20 = (y−4)(y−5) = 0, so y = 4 or y = 5. Comparing every combination — (3,4): x<y; (3,5): x<y; (4,4): x=y; (4,5): x<y — shows that x is never greater than y in any combination, so the only relationship that holds across all cases is x ≤ y.
5A.2 Time and Work — A Worked Example
A can complete a task in 20 days and B can complete it in 30 days. Working together with C, all three finish it in 10 days. To find how long C alone would take: the combined rate of A and B is 1/20 + 1/30 = 3/60 + 2/60 = 5/60 = 1/12 of the work per day. The combined rate of all three together is 1/10 per day. C's individual rate is therefore 1/10 − 1/12 = 6/60 − 5/60 = 1/60, meaning C alone would take 60 days to complete the task.
5A.3 Boats and Streams — A Worked Example
A boat travels downstream at an effective speed of 24 km/h and upstream at an effective speed of 16 km/h. The boat's speed in still water is the average of the two: (24+16)/2 = 20 km/h. The speed of the stream is half the difference: (24−16)/2 = 4 km/h. This pair of formulas — still-water speed equals the average of downstream and upstream speeds, and stream speed equals half their difference — should be applied directly whenever both downstream and upstream speeds are given.
5A.4 Mixtures and Alligation — A Worked Example
A mixture of milk and water is in the ratio 5:1, with a total volume of 36 litres. This means milk = 36×(5/6) = 30 litres and water = 36×(1/6) = 6 litres. If 4 litres of water is now added, the new water quantity is 6+4 = 10 litres, while milk remains 30 litres, giving a new ratio of 30:10, which simplifies to 3:1.
5A.5 Successive Percentage Changes — A Worked Example
An article's marked price is ₹2,000, and two successive discounts of 10% and 5% are applied. The selling price after both discounts is 2,000×0.90×0.95 = ₹1,710. Successive percentage discounts (or increases) must always be applied one after another to the running result, never added together as a single combined percentage (10%+5% = 15% would incorrectly give ₹1,700 instead of the correct ₹1,710).
5A.6 Compound Interest with Half-Yearly Compounding — A Worked Example
A sum of ₹8,000 is invested at 10% per annum for one year, compounded half-yearly. Since the compounding is half-yearly, the rate per period is 10%/2 = 5%, applied over 2 periods in one year. The amount is 8,000×(1.05)² = ₹8,820, giving a compound interest of ₹820 — noticeably more than the ₹800 that simple annual compounding at 10% for one year would give, because the interest itself starts earning interest after the first six months.
5A.7 Partnership — A Worked Example
A invests ₹30,000 for 12 months, and B invests ₹45,000 for 8 months. Since profits in a partnership are shared in the ratio of (capital × time) for each partner, A's share of the ratio is 30,000×12 = 3,60,000, and B's share is 45,000×8 = 3,60,000 — an equal ratio of 1:1. If the total profit is ₹20,000, each partner receives ₹10,000.
5A.8 Practice Set — Advanced Arithmetic & Algebra (16 MCQs)
- Given x² − 7x + 12 = 0 and y² − 9y + 20 = 0, what relationship between x and y holds for every valid combination of roots?
(a) x ≤ y (b) x ≥ y (c) x = y (d) x < y always strictly - A can complete a task in 20 days, B in 30 days. Working with C, all three finish it in 10 days. How long would C alone take?
(a) 60 days (b) 50 days (c) 45 days (d) 40 days - A boat's downstream speed is 24 km/h and upstream speed is 16 km/h. Find the boat's speed in still water.
(a) 20 km/h (b) 22 km/h (c) 18 km/h (d) 24 km/h - Using the same data as Q3, find the speed of the stream.
(a) 4 km/h (b) 5 km/h (c) 3 km/h (d) 6 km/h - A mixture of milk and water is in the ratio 5:1 with a total of 36 litres. If 4 litres of water is added, find the new ratio of milk to water.
(a) 3:1 (b) 5:2 (c) 4:1 (d) 2:1 - A marked price of ₹2,000 is discounted successively by 10% and 5%. Find the final selling price.
(a) ₹1,710 (b) ₹1,700 (c) ₹1,720 (d) ₹1,690 - Find the compound interest on ₹8,000 for 1 year at 10% per annum, compounded half-yearly.
(a) ₹820 (b) ₹800 (c) ₹840 (d) ₹810 - A invests ₹30,000 for 12 months and B invests ₹45,000 for 8 months. If the total profit is ₹20,000, find A's share.
(a) ₹10,000 (b) ₹12,000 (c) ₹8,000 (d) ₹11,000 - Given p² − 5p + 6 = 0 and q² − 7q + 12 = 0, what relationship holds between p and q across all valid root combinations?
(a) p ≤ q (b) p ≥ q (c) p = q (d) p > q always strictly - Pipe A fills a tank in 25 days and pipe B fills it in 20 days working alone; if both work with pipe C, all three together finish it in 10 days. How long would C alone take (assuming C also fills, not empties)?
(a) 100 days (b) 90 days (c) 110 days (d) 80 days - A boat's downstream speed is 30 km/h and its upstream speed is 18 km/h. Find the speed of the stream.
(a) 6 km/h (b) 8 km/h (c) 5 km/h (d) 7 km/h - A mixture of sugar solution is in the ratio 3:1 (sugar to water) totalling 40 litres. If 10 litres of water is added, find the new ratio of sugar to water.
(a) 3:2 (b) 2:1 (c) 5:2 (d) 4:3 - A price of ₹3,000 is increased successively by 20% and then decreased by 10%. Find the final price.
(a) ₹3,240 (b) ₹3,200 (c) ₹3,300 (d) ₹3,150 - Find the compound interest on ₹10,000 for 1 year at 20% per annum, compounded half-yearly.
(a) ₹2,100 (b) ₹2,000 (c) ₹2,200 (d) ₹2,050 - A invests ₹50,000 for 6 months and B invests ₹40,000 for 9 months. If the total profit is ₹22,000, find B's share.
(a) ₹12,000 (b) ₹10,500 (c) ₹9,000 (d) ₹13,000 - Given m² − 3m + 2 = 0 and n² − 6n + 8 = 0, what relationship holds between m and n across all valid root combinations?
(a) m ≤ n (b) m ≥ n (c) m = n (d) m > n always strictly
Answer Key with Explanations
1.(a) x ∈ {3,4} and y ∈ {4,5}; every combination gives x ≤ y (with equality only at x=4,y=4), so x ≤ y holds throughout.
2.(a) Combined A+B rate = 1/20+1/30 = 1/12; all three together = 1/10; C's rate = 1/10−1/12 = 1/60, so C alone takes 60 days.
3.(a) Still-water speed = (24+16)/2 = 20 km/h.
4.(a) Stream speed = (24−16)/2 = 4 km/h.
5.(a) Milk = 36×5/6 = 30 L; water = 36×1/6 = 6 L, plus 4 L added = 10 L; new ratio = 30:10 = 3:1.
6.(a) Selling price = 2,000×0.90×0.95 = ₹1,710.
7.(a) Amount = 8,000×(1.05)² = 8,820; compound interest = ₹820.
8.(a) A:B ratio = (30,000×12):(45,000×8) = 3,60,000:3,60,000 = 1:1, so A's share = 20,000/2 = ₹10,000.
9.(a) p ∈ {2,3} and q ∈ {3,4}; every combination gives p ≤ q, with equality only at p=3,q=3.
10.(a) Combined A+B rate = 1/25+1/20 = 4/100+5/100 = 9/100 per day; all three together = 1/10 = 10/100 per day; C's rate alone = 10/100−9/100 = 1/100 per day, so C alone takes 100 days.
11.(a) Stream speed = (30−18)/2 = 6 km/h.
12.(a) Sugar = 40×3/4 = 30 L; water = 40×1/4 = 10 L, plus 10 L added = 20 L; new ratio = 30:20 = 3:2.
13.(a) Price after a 20% increase = 3,000×1.20 = 3,600; after a 10% decrease = 3,600×0.90 = ₹3,240.
14.(a) Amount = 10,000×(1.10)² = 12,100; compound interest = ₹2,100.
15.(a) A:B ratio = (50,000×6):(40,000×9) = 3,00,000:3,60,000 = 5:6; total 11 parts = 22,000, so each part = 2,000, giving B's share = 6×2,000 = ₹12,000.
16.(a) m ∈ {1,2} and n ∈ {2,4}; every combination gives m ≤ n, with equality only at m=2,n=2.