What is HCF, LCM, Average, Age Problems & Chain Rule in MAT?

HCF, LCM, Average, Age Problems & Chain Rule is a topic in the Mathematical Skills syllabus of MAT. It carries a weight of 3% in the exam. Our notes cover the key concepts, formulas, and problem-solving approaches you need to master this topic.

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HCF, LCM, Average, Age Problems & Chain Rule

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

HCF (Highest Common Factor) & LCM (Least Common Multiple)

HCF (also called GCD — Greatest Common Divisor) is the largest number that divides two or more numbers exactly. LCM is the smallest number that is a multiple of both numbers.

Key relationship: For any two numbers a and b: $$a \times b = \text{HCF}(a,b) \times \text{LCM}(a,b)$$

HCF shortcut methods:

Prime factorisation: Take the LOWEST power of each common prime
Division method: Keep dividing by the smallest prime until remainder is 0

LCM shortcut methods:

Prime factorisation: Take the HIGHEST power of each prime appearing in either number
For two fractions: $\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}$

Quick HCF/LCM table:

Numbers	HCF	LCM
12, 15	3	60
18, 24	6	72
24, 36	12	72
35, 40	5	280

Averages

Average = $\frac{\text{Sum of observations}}{\text{Number of observations}}$

Weighted Average: If group 1 has average $a_1$ and $n_1$ items; group 2 has average $a_2$ and $n_2$ items: $$\text{Combined Average} = \frac{n_1 a_1 + n_2 a_2}{n_1 + n_2}$$

Average tricks for MAT:

If each term increases by same value x: new average = old average + x
Adding a number equal to current average doesn’t change the average

⚡ Exam tip for MAT: In questions asking “replace a number” — use: New number = Old number ± n × change_in_average (where n = total count)

Age Problems — Key Relations

Situation	Equation
Person is n years older	Age now: x + n
Ratio given (e.g., 3:5)	Let ages be 3x and 5x
n years ago	Subtract n from each
n years later	Add n to each

⚡ Exam tip for MAT: Age problems are easiest with algebra — assign present age as x. Remember: ratio questions multiply the ratio value by k, not x directly.

Chain Rule

Chain rule is used when dealing with DIRECT or INVERSE proportions.

Direct proportion: If more work → more workers (or more days), and vice versa. Formula: $M_1 \times D_1 \times W_2 = M_2 \times D_2 \times W_1$

Where M = workers, D = days, W = work units

Standard chain rule formula: $$\frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2}$$

⚡ Exam tip for MAT: In chain rule, all quantities in one column must vary DIRECTLY with each other. If time decreases when efficiency increases → it’s an INVERSE proportion problem.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

HCF and LCM — Deep Dive

Finding HCF by Prime Factorisation

Example: Find HCF of 84 and 144.

$84 = 2^2 \times 3 \times 7$
$144 = 2^4 \times 3^2$
Common factors: $2^2$ and $3^1$
HCF = $2^2 \times 3 = 12$

Finding LCM by Prime Factorisation

Example: Find LCM of 12, 18, 27

$12 = 2^2 \times 3^1$
$18 = 2^1 \times 3^2$
$27 = 3^3$
LCM = $2^2 \times 3^3 = 4 \times 27 = 108$

HCF of Fractions

$$\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}$$ Example: HCF of $\frac{3}{4}$ and $\frac{5}{6}$ = $\frac{\text{HCF}(3,5)}{\text{LCM}(4,6)} = \frac{1}{12}$

LCM of Fractions

$$\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}$$ Example: LCM of $\frac{3}{4}$ and $\frac{5}{6}$ = $\frac{\text{LCM}(3,5)}{\text{HCF}(4,6)} = \frac{15}{2} = 7.5$

Word Problems on HCF & LCM

Q: Two bells ring together every 12 and 15 minutes respectively. After how many minutes will they ring together again?

LCM of 12 and 15 = 60 minutes = 1 hour

Q: Find the largest number that divides 204 and 870, leaving remainders 5 and 8 respectively.

204 − 5 = 199; 870 − 8 = 862
HCF of 199 and 862 = 199

Average — Advanced Applications

Finding Missing Number

If average of n numbers is A, and one number is replaced: $$\text{New number} = \text{Old number} + n \times (\text{New average} - \text{Old average})$$

Q: Average of 15 numbers is 22. One number 42 is replaced by 18. Find new average.

Sum of all = 15 × 22 = 330
New sum = 330 − 42 + 18 = 306
New average = 306/15 = 20.4

Average of First n Natural Numbers

$$\text{Average} = \frac{n+1}{2}$$

Q: Average of first 20 even numbers?

Even numbers: 2, 4, 6, …, 40
Average = (2 + 40)/2 = 21

Age Problems — Worked Examples

Q: Ratio of ages of A and B is 3:5. After 6 years, the ratio becomes 5:7. Find their present ages.

Let present ages be 3x and 5x
After 6 years: (3x+6)/(5x+6) = 5/7
7(3x+6) = 5(5x+6)
21x + 42 = 25x + 30
4x = 12 → x = 3
A = 9 years, B = 15 years ✓

Q: A father is 4 times as old as his son. In 20 years, he will be twice as old. Find their present ages.

Let son’s age = x, father’s age = 4x
4x + 20 = 2(x + 20)
4x + 20 = 2x + 40
2x = 20 → x = 10
Son = 10 years, Father = 40 years

Chain Rule — Direct & Inverse Proportion

Type	Relationship	Formula
Direct	More workers → More work done	$M_1D_1W_2 = M_2D_2W_1$
Inverse	More workers → Less time	$M_1D_1 = M_2D_2$

Q: 12 workers can complete a job in 20 days. How many days will 15 workers take?

Workers increase → days decrease (INVERSE)
$12 \times 20 = 15 \times D_2$
$D_2 = 240/15 = 16$ days

Q: If 8 machines produce 200 units in 5 hours, how many units will 12 machines produce in 8 hours?

$M_1 = 8, H_1 = 5, W_1 = 200$
$M_2 = 12, H_2 = 8, W_2 = ?$
$8 \times 5 / 200 = 12 \times 8 / W_2$
$W_2 = 12 \times 8 \times 200 / (8 \times 5) = 12 \times 8 \times 200 / 40 = 480$ units

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Advanced HCF & LCM — Coprime Numbers

Two numbers are coprime (or relatively prime) if their HCF = 1. Note: coprime numbers don’t have to be prime themselves — e.g., 8 and 9 are coprime (HCF = 1).

Properties:

If a and b are coprime: LCM(a,b) = a × b
If p is prime and p doesn’t divide a: HCF(p,a) = 1
Product of two numbers = Product of their HCF and LCM (always true)

Q: Two numbers have HCF 12 and LCM 360. One number is 48. Find the other.

$a \times b = \text{HCF} \times \text{LCM} = 12 \times 360 = 4320$
$b = 4320 / 48 = 90$

Q: Find the greatest number that divides 1365, 1865, and 2765 leaving the same remainder.

Subtract the smallest from each: 1865−1365=500, 2765−1365=1400
HCF of 500 and 1400 = 100

Average of Grouped Data

For data with frequencies: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

Q: Find average of:

Marks	Frequency
10	3
20	5
30	7
40	3

$\sum f_i x_i = 30 + 100 + 210 + 120 = 460$
$\sum f_i = 3 + 5 + 7 + 3 = 18$
Average = 460/18 = 25.56

Moving Average (used in time-series analysis):

3-year moving average: Average of consecutive 3-year periods
Used to smooth out seasonal fluctuations in data

Age Problems — Complex Scenarios

Q: The sum of ages of 4 children born at intervals of 3 years each is 50 years. Find the age of the youngest.

Let youngest age = x, then ages: x, x+3, x+6, x+9
x + (x+3) + (x+6) + (x+9) = 50
4x + 18 = 50 → 4x = 32 → x = 8 years

Q: Ten years ago, mother was 4 times as old as her daughter. The sum of their ages now is 70. Find their present ages.

10 years ago: daughter = d−10, mother = 4(d−10)
Now: mother = 4(d−10)+10 = 4d−30
d + (4d−30) = 70 → 5d = 100 → d = 20
Daughter = 20 years, Mother = 50 years

⚡ Common mistake: Don’t assume “sum of ages is constant” — it isn’t! Only the difference in ages stays constant over time.

Chain Rule — Complex Work Problems

Q: 20 men can build a wall 200m long in 30 days. How many men are needed to build 300m in 15 days?

$W_1 = 200, M_1 = 20, D_1 = 30$
$W_2 = 300, M_2 = ?, D_2 = 15$
$M_2 = \frac{200 \times 30 \times 300}{300 \times 15 \times 200} \times 20$ — Wait, formula: $M_1 D_1 / W_1 = M_2 D_2 / W_2$
$M_2 = 20 \times 30 \times 300 / (200 \times 15) = 20 \times 30 \times 300 / 3000 = 60$ men

Q: A pump fills a tank in 2 hours but a leak can empty it in 6 hours. How long to fill with leak active?

Pump rate = 1/2 tank/hour
Leak rate = 1/6 tank/hour (negative)
Net rate = 1/2 − 1/6 = 1/3 tank/hour
Time = 3 hours

⚡ MAT exam shortcut: For HCF-LCM of three numbers, always check if a number is the HCF/LCM of the other two before computing.

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