Ratio, Proportion, Partnership and Alligation

• A ratio a : b is the fraction a/b. Multiplying or dividing both terms by the same non-zero number does not change it (a : b = ka : kb).

• The ratio compares quantities with the same unit; convert units first (for example, 2 hours : 40 minutes = 120 : 40 = 3 : 1).

• Compound ratio of a : b and c : d is ac : bd.

• Duplicate ratio of a : b is a squared : b squared; triplicate is a cubed : b cubed; sub-duplicate is the square roots; sub-triplicate is the cube roots.

• Inverse ratio of a : b is b : a.

• To compare ratios, convert each to a fraction or decimal.

• To divide a quantity N in the ratio a : b : c, the shares are N x a/(a + b + c), N x b/(a + b + c) and N x c/(a + b + c).

If A : B = 2 : 3 and B : C = 4 : 5, make the common term equal: B becomes 3 x 4 = 12 in the first and 4 x 3 = 12 in the second. Then A : B : C = (2 x 4) : 12 : (5 x 3) = 8 : 12 : 15.

• Direct variation: two quantities rise or fall together, and their ratio stays constant. If 12 pens cost 90, 20 pens cost 150.

• Inverse variation: one rises when the other falls, and their product stays constant. If 6 workers finish a job in 15 days, 9 workers finish it in 6 x 15/9 = 10 days.

• Chain rule (compound proportion): (men x days x hours per day)/work is the same in both cases. If 10 men finish a job in 12 days, 15 men working the same hours take 10 x 12/15 = 8 days.

• Profit is shared in the ratio of capital x time for each partner.

• Same time: ratio of capitals. Same capital: ratio of times.

• A working partner first receives a fixed commission; the rest is split by the ratio.

• If capital changes during the year, add up capital x months for each period.

Example. A invests 5000 for 12 months; B invests 6000 for 8 months. The profit is 4500.

• A : B = (5000 x 12) : (6000 x 8) = 60000 : 48000 = 5 : 4.

• A gets 4500 x 5/9 = 2500; B gets 4500 x 4/9 = 2000.

Alligation rule. If a cheaper item at price c and a dearer item at price d are mixed to give a mean price m:

(quantity of cheaper) : (quantity of dearer) = (d - m) : (m - c)

Example. Rice at 40 per kg is mixed with rice at 60 per kg to make a mixture worth 52 per kg.

• Cheaper : dearer = (60 - 52) : (52 - 40) = 8 : 12 = 2 : 3.

Milk and water example. A 40-litre mixture has milk and water in the ratio 3 : 1. How much water must be added to make the ratio 1 : 1?

• Milk = 30 L, water = 10 L. For a 1 : 1 ratio the water must also be 30 L, so add 20 L.

Replacement formula. A vessel holds x litres of pure liquid. y litres are removed and replaced by water, and this is done n times in total.

• Pure liquid left = x (1 - y/x)^n.

• Example: 40 L of milk, 4 L replaced twice. Milk left = 40 x (0.9) squared = 40 x 0.81 = 32.4 L.

Example 1. Divide 1050 among A, B and C in the ratio 2 : 3 : 5.

• Total parts = 10, one part = 105. Shares are 210, 315 and 525.

Example 2. The ratio of two numbers is 3 : 4 and their sum is 98. Find the numbers.

• One part = 98/7 = 14. The numbers are 42 and 56.

Example 3. If A is 25% of B, what is A : B?

• A = B/4, so A : B = 1 : 4.

Example 4. The mean proportional between 4 and 36 is the square root of (4 x 36) = square root of 144 = 12.

• The k-method: write the quantities as ka, kb, kc and use the given sum or difference to find k.

• Percentage to ratio: convert percentages to simple fractions (20% = 1/5, 25% = 1/4) and compare.

• Alligation shortcut: draw the cross: the mean in the centre, cheaper and dearer prices at the left corners, and subtract diagonally.

• Equal ratio test: if cross-products are equal, the ratios are equal.

• Mixing up the cheaper-to-dearer order in alligation. The quantity of the cheaper item corresponds to the difference taken from the dearer price.

• Adding ratios directly. Ratios must be converted to a common base before combining.

• Using direct proportion when the situation is inverse (workers and days).