Episode 8 — Aptitude and Reasoning / 8.5 — Ratio and Proportion
8.5.b Tips, Tricks, and Shortcuts -- Ratio and Proportion
1. Quick Ratio Comparison -- Cross Multiplication
To compare a/b and c/d without finding a common denominator:
Cross multiply:
a x d vs b x c
If a*d > b*c => a/b > c/d
If a*d < b*c => a/b < c/d
If a*d = b*c => a/b = c/d
Example:
Compare 7/11 and 5/8.
7 x 8 = 56
11 x 5 = 55
56 > 55 => 7/11 > 5/8
This takes about 3 seconds vs. converting to common denominators.
2. The "k-Multiplier" Technique
When a ratio is given as a : b, immediately write the actual quantities as ak and bk. This converts ratio problems into simple algebra.
If A : B = 3 : 5 => A = 3k, B = 5k
Now use any extra condition to find k.
Example:
A : B = 3 : 5 and A + B = 120.
3k + 5k = 120 => 8k = 120 => k = 15
A = 45, B = 75
This works for any number of terms:
A : B : C = 2 : 3 : 7 and total = 360
2k + 3k + 7k = 360 => 12k = 360 => k = 30
A = 60, B = 90, C = 210
3. Fraction-to-Ratio Shortcut
When a ratio involves fractions, multiply all terms by the LCM of the denominators.
(1/2) : (2/3) : (3/4)
LCM of 2, 3, 4 = 12
= (1/2 x 12) : (2/3 x 12) : (3/4 x 12)
= 6 : 8 : 9
Speed tip: This is much faster than converting each fraction to a common denominator first.
4. Scaling Trick for Combining Ratios
When combining A:B and B:C, make B the same value in both ratios by finding the LCM.
A : B = 2 : 3 B : C = 4 : 5
B values are 3 and 4. LCM = 12.
A : B = 2:3 => x4 => 8 : 12
B : C = 4:5 => x3 => 12 : 15
Therefore A : B : C = 8 : 12 : 15
Extended version (three pairs):
A : B = 2 : 3, B : C = 4 : 5, C : D = 6 : 7
Step 1: Combine A:B and B:C first.
B = LCM(3,4) = 12
A:B = 8:12, B:C = 12:15
A:B:C = 8:12:15
Step 2: Combine with C:D.
C = LCM(15,6) = 30
A:B:C = 8:12:15 => x2 => 16:24:30
C:D = 6:7 => x5 => 30:35
A:B:C:D = 16:24:30:35
5. The Alligation Cross Method (Visual Shortcut)
This is the fastest method for mixture problems. Draw the cross diagram:
Step 1: Write the two values on the left and right, with the mean in the middle.
Cheaper (C) Dearer (D)
\ /
\ /
Mean value (M)
/ \
/ \
|D - M| |M - C|
Step 2: The ratio is:
Cheaper : Dearer = |D - M| : |M - C|
Example: Mixing Rs 30/kg tea with Rs 50/kg tea to get Rs 35/kg mixture.
30 50
\ /
\ /
35
/ \
/ \
|50-35| |35-30|
= 15 = 5
Ratio = 15 : 5 = 3 : 1
Alligation for "Replacement" Problems
When a portion is removed and replaced with another liquid:
After n operations of removing x litres from a container of C litres:
Quantity of original liquid = C x (1 - x/C)^n
Example:
A container has 80 litres of milk. 8 litres are removed and
replaced with water. This is done 3 times. Find milk remaining.
Milk = 80 x (1 - 8/80)^3
= 80 x (72/80)^3
= 80 x (9/10)^3
= 80 x 729/1000
= 58.32 litres
6. Ratio-Percentage Quick Conversions
Memorize these common conversions for speed:
+------------+-----------------+-------------------+
| Ratio a:b | a as % of (a+b) | b as % of (a+b) |
+------------+-----------------+-------------------+
| 1 : 1 | 50% | 50% |
| 1 : 2 | 33.33% | 66.67% |
| 1 : 3 | 25% | 75% |
| 1 : 4 | 20% | 80% |
| 2 : 3 | 40% | 60% |
| 3 : 4 | 42.86% | 57.14% |
| 3 : 5 | 37.5% | 62.5% |
| 3 : 7 | 30% | 70% |
+------------+-----------------+-------------------+
Converting a Percentage to a Ratio
A is 25% more than B.
=> A/B = 125/100 = 5/4
=> A : B = 5 : 4
A is 20% less than B.
=> A/B = 80/100 = 4/5
=> A : B = 4 : 5
Quick Percentage-Ratio Table
+---------------+--------+
| % more/less | Ratio |
+---------------+--------+
| 10% more | 11:10 |
| 20% more | 6:5 |
| 25% more | 5:4 |
| 33.33% more | 4:3 |
| 50% more | 3:2 |
| 100% more | 2:1 |
| 10% less | 9:10 |
| 20% less | 4:5 |
| 25% less | 3:4 |
| 33.33% less | 2:3 |
| 50% less | 1:2 |
+---------------+--------+
7. Income-Expenditure-Savings Shortcut
Income ratio = a : b
Expenditure ratio = c : d
Each saves S.
Incomes: ax, bx Expenditures: cy, dy
ax - cy = S ...(i)
bx - dy = S ...(ii)
Subtract (ii) from (i):
(a - b)x = (c - d)y
=> x/y = (c - d)/(a - b)
This gives you the relationship between x and y directly.
8. Partnership Shortcut
When the profit/loss is to be divided and you know each partner's capital and time:
Share of A = (C_A x T_A) / (Sum of all C x T) x Total Profit
Shortcut for equal time: Just use the capital ratio directly.
Shortcut for equal capital: Just use the time ratio directly.
9. Age Ratio Shortcut
If present age ratio = a : b and ratio after t years = c : d:
Let present ages = ak and bk.
(ak + t) / (bk + t) = c/d
Solve for k:
d(ak + t) = c(bk + t)
adk + dt = bck + ct
k(ad - bc) = ct - dt = t(c - d)
k = t(c - d) / (ad - bc)
Direct formula:
k = t(c - d) / (ad - bc)
Example: Present ratio = 4:5, after 6 years = 5:6.
k = 6(5-6) / (4x6 - 5x5) = 6(-1) / (24-25) = -6 / -1 = 6
Ages: 4(6) = 24 and 5(6) = 30.
For "t years ago" problems, replace t with -t:
Present ratio = 4:5, ratio 4 years ago = 3:4.
k = (-4)(3-4) / (4x4 - 5x3) = (-4)(-1) / (16-15) = 4/1 = 4
Ages: 16 and 20.
Verify 4 years ago: 12:16 = 3:4 (correct)
10. Componendo-Dividendo Speed Trick
If you see an equation of the form:
(x + y) / (x - y) = p/q
Apply componendo-dividendo instantly:
x/y = (p + q) / (p - q)
Example:
(3a + 4b) / (3a - 4b) = 7/3
Applying C-D:
3a / 4b = (7 + 3) / (7 - 3) = 10/4 = 5/2
=> a/b = (5 x 4) / (2 x 3) = 20/6 = 10/3
=> a : b = 10 : 3
11. Common Exam Patterns to Recognize Instantly
Pattern 1: "A quantity is divided into two parts"
=> Use the ratio division formula: Part = Total x (ratio term / sum of ratio terms)
Pattern 2: "The ratio becomes p:q when x is added/subtracted"
=> Write as (a*k + x) / (b*k + x) = p/q and solve for k.
(Use - x for subtraction.)
Pattern 3: "Incomes in ratio a:b, expenses in ratio c:d, each saves S"
=> Set up two equations with two unknowns (x and y) and solve.
Pattern 4: "Mixture of two things at different prices to get a target price"
=> Alligation. Draw the cross diagram.
Pattern 5: "Partners invest different amounts for different times"
=> Capital x Time ratio for profit sharing.
Pattern 6: "Speed ratio is a:b, find time ratio"
=> Time ratio = b:a (inverse of speed ratio, for same distance).
Pattern 7: "If a:b = 2:3, b:c = 4:5, find a:c"
=> Combine via LCM on the common term.
Or directly: a/c = (a/b) x (b/c) = (2/3) x (4/5) = 8/15
So a:c = 8:15
12. Avoid These Common Mistakes
1. UNITS: Always convert to the same unit before forming a ratio.
Rs 5 : 50 paise => 500 paise : 50 paise = 10 : 1 (NOT 5 : 50)
2. ORDER: a:b is NOT the same as b:a.
"Ratio of boys to girls" = Boys/Girls.
3. ADDING TO RATIOS: If A:B = 3:4, after adding 5 to each,
the ratio is NOT 3:4. It becomes (3k+5):(4k+5).
4. FRACTION RATIOS: Don't forget to clear fractions.
(2/3) : (4/5) is NOT 2:4.
5. PROPORTION SETUP: In proportion a:b::c:d, make sure the
corresponding quantities are in the right positions.
6. ALLIGATION: The mean value must lie BETWEEN the two given values.
If it doesn't, check your setup.
13. Mental Math Shortcuts for Ratios
Dividing amounts mentally
To divide 720 in ratio 2:3:4:
Sum = 9
Think: 720/9 = 80 (this is the "unit")
Parts: 2x80=160, 3x80=240, 4x80=320
Quick simplification
Look for common factors quickly:
144 : 180 => both divisible by 36 => 4 : 5
(or divide by 12 first: 12:15, then by 3: 4:5)
Ratio from percentage
A is what % of B?
If A:B = 3:5, then A is (3/5)x100 = 60% of B.
And B is (5/3)x100 = 166.67% of A.