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Module 2: S4 Quant

Topic 8: Percentages, Ratios, Averages & Mixtures

1 Percentage Hacks (The Foundation of Quant)

Percentages are the lifeblood of administrative arithmetic. Do not calculate manually. Use the fundamental theorems of proportional shifts.

1. The "Reversal" Theorem

If $X\%$ of $Y$ is too hard to calculate, simply flip it. $X\% \text{ of } Y = Y\% \text{ of } X$.

Problem: Find $16\%$ of $25$.
Flip it: Find $25\%$ of $16$.
Logic: $25\%$ is $\frac{1}{4}$.
$\frac{1}{4} \times 16 = \mathbf{4}$.

2. The "More/Less" Balance Trap

If the Principal's salary is $25\%$ MORE than the Teacher's, the Teacher's salary is NOT $25\%$ LESS than the Principal's! The base has changed.

Universal Formula:
If A is $x\%$ more than B, B is less than A by:
$$ \left( \frac{x}{100 + x} \times 100 \right)\% $$ Solution: $\frac{25}{125} \times 100 = \frac{1}{5} \times 100 = \mathbf{20\%}$.

3. Net Effective Percentage (Successive Change)

Used when a school's budget is increased by $A\%$ and then decreased by $B\%$. Never just add/subtract the percentages! Use the Successive Formula:

$$ \pm A \pm B + \frac{(\pm A)(\pm B)}{100} $$

Example: Budget increased by $20\%$ ($+20$), then cut by $10\%$ ($-10$).
Net $= 20 - 10 + \frac{20 \times (-10)}{100} = 10 - 2 = \mathbf{+8\%}$ (Net Increase of 8%).

2 Ratio & Proportion (Combining & Distributing)

The "Z-Pattern" for Combining Ratios

If the ratio of Science to Arts students is $A:B$, and Arts to Commerce is $C:D$, how do you find Science:Arts:Commerce ($A:B:C$)?

A : B C : D (A×C) : (B×C) : (B×D)

Mean, Third & Fourth Proportional

If $a, b, c, d$ are in proportion, then $\frac{a}{b} = \frac{c}{d}$ (or $a \times d = b \times c$).

Mean Proportional of $a$ and $b$:
$$ x = \sqrt{a \times b} $$ Third Proportional of $a$ and $b$:
$$ x = \frac{b^2}{a} $$ Fourth Proportional of $a, b, c$:
$$ x = \frac{b \times c}{a} $$

3 Averages (Replacement & Group Shifts)

The "New Arrival" Trick

When a new teacher joins the staff, or a student leaves, do NOT recalculate the total sum. Use the Deviation Formula.

Formula:
$$\text{New Value} = \text{Old Avg} \pm (\text{Change in Avg} \times \text{New Total Count})$$
Use $(+)$ if joining increases avg. Use $(-)$ if joining decreases avg.

Admin Example:

The average age of 24 students in a class is 15 years. When the class teacher is included, the average age increases by 1 year. Find the age of the teacher.

Old Avg = 15
Change = +1
New Total Count = 24 + 1 = 25
$$\text{Teacher's Age} = 15 + (1 \times 25)$$ $$\text{Teacher's Age} = 15 + 25 = \mathbf{40 \text{ years}}.$$

4 Mixtures & Alligation (The Ultimate Time-Saver)

Alligation is not just for liquids. It is a magical technique used to find the ratio in which two groups (with different averages, prices, or percentages) are mixed to form a combined group.

The Alligation "X" Diagram

Cheaper ($C$) Dearer ($D$) Mean ($M$) $D - M$ Ratio (:) $M - C$

Admin Context (Pass Ratio):

In a school, the average marks of girls in an exam is 73 and that of boys is 71. The average marks of the entire school is 71.8. Find the percentage of girls in the school.

Step 1: Set up Alligation
Girls (73)           Boys (71)
          \        /
            (71.8)
          /        \
(71.8 - 71)       (73 - 71.8)
Ratio: $0.8 : 1.2$
Simplify: Multiply by 10 ➔ $8 : 12 \implies \mathbf{2 : 3}$

Step 2: Find Percentage
Total parts = $2 + 3 = 5$. Girls = 2 parts.
$$\frac{2}{5} \times 100 = \mathbf{40\% \text{ Girls}}.$$