What a ratio actually says
a : b tells you how the two quantities compare, not how big they are. 2 : 3 could be 20 and 30, or 200 and 300. That missing size is exactly what you solve for.
The k-method. Write the parts with a common multiplier. If two numbers are in the ratio 5 : 7, write them as 5k and 7k. Every ratio problem becomes one equation in k. Learn this and the whole topic collapses into routine algebra.
Dividing an amount in a ratio
To split N in the ratio a : b : c, add the parts, divide, multiply back.
- Total parts = a + b + c
- One part = N / (a + b + c)
- Shares = a parts, b parts, c parts
Proportion
a : b :: c : d means a/b = c/d, so a*d = b*c. Cross multiply.
The mean proportional between a and b is the square root of a*b. The mean proportional between 4 and 9 is 6.
Compounded ratios. If A : B = 2 : 3 and B : C = 4 : 5, make B match. B is 3 in the first and 4 in the second, so scale to 12: multiply the first by 4 and the second by 3. A : B = 8 : 12 and B : C = 12 : 15, giving A : B : C = 8 : 12 : 15.
You cannot add ratios. If A : B = 2 : 3 and C : D = 4 : 5, then (A+C) : (B+D) is not 6 : 8. Ratios only combine through a shared term, and only after that term is made equal in both.
Partnership
Two friends open a shop. Profit is shared in proportion to how much money each put in and for how long it stayed in.
Profit share is proportional to capital multiplied by time.
A partner who invests 30,000 for 12 months has 360,000 rupee-months. A partner who invests 60,000 for 6 months also has 360,000. They split the profit equally, even though one invested twice as much.
Two words from the syllabus. A working partner does the running of the business and may take a salary or a percentage off the top before the remaining profit is divided by capital. A sleeping partner only puts in money.
Worked problem 1
Divide Rs 1,050 among A, B and C in the ratio 2 : 3 : 5.
Total parts = 2 + 3 + 5 = 10
One part = 1050 / 10 = 105
A = 2 * 105 = Rs 210
B = 3 * 105 = Rs 315
C = 5 * 105 = Rs 525
Check: 210 + 315 + 525 = 1050. Correct.
Worked problem 2
Two numbers are in the ratio 5 : 7. If 6 is added to each, the ratio becomes 3 : 4. Find the numbers.
Write them as 5k and 7k.
(5k + 6) / (7k + 6) = 3 / 4
Cross multiply:
4 (5k + 6) = 3 (7k + 6)
20k + 24 = 21k + 18
24 - 18 = 21k - 20k
6 = k
Numbers = 5 6 = 30 and 7 6 = 42
Check: (30 + 6) / (42 + 6) = 36 / 48 = 3/4. Correct.
Worked problem 3
A starts a business with Rs 24,000. After some months B joins with Rs 36,000 and stays for 9 months. At the end of the year the total profit is Rs 34,000. Find each partner's share, given that A was in for all 12 months.
Work in thousands to keep the numbers small.
A's capital-months = 24 * 12 = 288
B's capital-months = 36 * 9 = 324
Ratio A : B = 288 : 324
Divide both by 36: = 8 : 9
Total parts = 8 + 9 = 17
One part = 34000 / 17 = 2000
A's share = 8 * 2000 = Rs 16,000
B's share = 9 * 2000 = Rs 18,000
Check: 16000 + 18000 = 34000. Correct.
Always reduce the capital-months ratio before dividing the profit. 288 : 324 reduces to 8 : 9, and dividing 34,000 by 17 is easy. Dividing by 612 is not.
Common traps
- Time in months for one partner and years for another. Convert first.
- A partner who withdraws half the capital midway. Split that partner into two blocks and add the capital-months.
- The question asks for the difference between two shares, not a share. Read the last line again before you circle an option.
Next: averages, which is a ratio question with the parts hidden.