HCF and LCM of Fractions (பின்னங்களின் மீ.பொ.வ மற்றும் மீ.சி.ம)

Overview

HCF and LCM of Fractions (பின்னங்களின் மீ.பொ.வ மற்றும் மீ.சி.ம)

This section covers problems related to finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) of fractions. These are common questions in aptitude tests.

Key Formulas (முக்கிய சூத்திரங்கள்)

To solve these problems, use the following formulas:

1. HCF of Fractions (பின்னங்களின் மீ.பொ.வ):

   $$\text{HCF} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$ $$\text{பின்னங்களின் மீ.பொ.வ} = \frac{\text{தொகுதிகளின் மீ.பொ.வ}}{\text{பகுதிகளின் மீ.சி.ம}}$$

2. LCM of Fractions (பின்னங்களின் மீ.சி.ம):

   $$\text{LCM} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$ $$\text{பின்னங்களின் மீ.சி.ம} = \frac{\text{தொகுதிகளின் மீ.சி.ம}}{\text{பகுதிகளின் மீ.பொ.வ}}$$

Example Problem (எடுத்துக்காட்டு கணக்கு)

Question: Find the HCF of $\frac{3}{5}, \frac{6}{10}, \frac{9}{20}$.

Solution:

1. Identify the formula: To find the HCF of fractions, we use: $HCF = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$

2. Find HCF of Numerators (தொகுதிகளின் மீ.பொ.வ): The numerators are 3, 6, and 9.

   3 | 3, 6, 9
   --|--------
     | 1, 2, 3

   The HCF of (3, 6, 9) is 3.

3. Find LCM of Denominators (பகுதிகளின் மீ.சி.ம): The denominators are 5, 10, and 20.

   5 | 5, 10, 20
   --|----------
   2 | 1,  2,  4
   --|----------
     | 1,  1,  2

   The LCM of (5, 10, 20) is $5 \times 2 \times 2 = \textbf{20}$.

4. Calculate the final HCF:

   $$HCF = \frac{3}{20}$$

---

1. $\frac{4}{9}, \frac{2}{5}, \frac{6}{8}, \frac{2}{5}$ இவற்றின் HCF காண்க.

   • a) $\frac{1}{180}$
   • b) $\frac{1}{481}$
   • c) $\frac{1}{350}$
   • d) $\frac{1}{142}$

   Answer: a) $\frac{1}{180}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (4, 2, 6, 2):

   $$HCF(4, 2, 6, 2) = 2$$

   LCM of Denominators (9, 5, 8, 5):

   $$\begin{align*} LCM(9, 5, 8, 5) &= 5 \times 9 \times 8 \\ &= 360 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{2}{360} = \frac{1}{180}$$
   Why this question belongs to HCF of Fractions

   This question asks to find the HCF (மீ.பொ.வ) of a set of fractions, which directly applies the formula for HCF of fractions.

2. $\frac{3}{8}, \frac{2}{9}, \frac{3}{5}, \frac{1}{3}$ இவற்றின் HCF காண்க.

   • a) $\frac{1}{45}$
   • b) $\frac{1}{90}$
   • c) $\frac{1}{180}$
   • d) $\frac{1}{360}$

   Answer: d) $\frac{1}{360}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (3, 2, 3, 1):

   $$HCF(3, 2, 3, 1) = 1$$

   LCM of Denominators (8, 9, 5, 3):

   $$\begin{align*} LCM(8, 9, 5, 3) &= 3 \times 8 \times 3 \times 5 \\ &= 360 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{1}{360}$$
   Why this question belongs to HCF of Fractions

   This question involves finding the HCF for given fractions, using keywords like 'HCF' (மீ.பொ.வ) and 'fraction' (பின்னம்).

3. $\frac{4}{9}, \frac{8}{27}, \frac{6}{45}$ இவற்றின் HCF காண்க.

   • a) $\frac{4}{135}$
   • b) $\frac{2}{135}$
   • c) $\frac{2}{45}$
   • d) $\frac{4}{45}$

   Answer: b) $\frac{2}{135}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (4, 8, 6):

   $$HCF(4, 8, 6) = 2$$

   LCM of Denominators (9, 27, 45):

   $$\begin{align*} LCM(9, 27, 45) &= 3 \times 3 \times 3 \times 5 \\ &= 135 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{2}{135}$$
   Why this question belongs to HCF of Fractions

   The problem requires calculating the HCF of fractions, a standard problem in this category.

4. $\frac{4}{9}, \frac{10}{21}, \frac{20}{63}$ இவற்றின் HCF காண்க.

   • a) $\frac{2}{21}$
   • b) $\frac{2}{31}$
   • c) $\frac{2}{63}$
   • d) $\frac{2}{189}$

   Answer: c) $\frac{2}{63}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (4, 10, 20):

   $$HCF(4, 10, 20) = 2$$

   LCM of Denominators (9, 21, 63):

   $$\begin{align*} LCM(9, 21, 63) &= 3 \times 3 \times 7 \\ &= 63 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{2}{63}$$
   Why this question belongs to HCF of Fractions

   This question asks for the HCF (மீ.பொ.வ) of several fractions, which is the core concept of this topic.

5. $\frac{2}{3}, \frac{4}{9}, \frac{5}{6}, \frac{7}{12}$ இவற்றின் HCF காண்க.

   • a) $\frac{1}{3}$
   • b) $\frac{1}{46}$
   • c) $\frac{1}{12}$
   • d) $\frac{1}{36}$

   Answer: d) $\frac{1}{36}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (2, 4, 5, 7):

   $$HCF(2, 4, 5, 7) = 1$$

   LCM of Denominators (3, 9, 6, 12):

   $$\begin{align*} LCM(3, 9, 6, 12) &= 3 \times 2 \times 3 \times 2 \\ &= 36 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{1}{36}$$
   Why this question belongs to HCF of Fractions

   This question involves keywords like HCF (மீ.பொ.வ) and fractions (பின்னங்கள்), which are common in HCF of Fractions problems.

6. $\frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}$ இவற்றின் HCF காண்க.

   • a) $\frac{1}{24}$
   • b) $\frac{1}{48}$
   • c) $\frac{2}{25}$
   • d) $\frac{7}{49}$

   Answer: a) $\frac{1}{24}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (1, 3, 5, 7):

   $$HCF(1, 3, 5, 7) = 1$$

   LCM of Denominators (2, 4, 6, 8):

   $$\begin{align*} LCM(2, 4, 6, 8) &= 2 \times 2 \times 3 \times 2 \\ &= 24 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{1}{24}$$
   Why this question belongs to HCF of Fractions

   This is a direct application of the formula for finding the HCF of fractions.

7. $\frac{3}{5}, \frac{6}{10}, \frac{9}{20}$ இவற்றின் HCF காண்க.

   • a) $\frac{3}{16}$
   • b) $\frac{6}{5}$
   • c) $\frac{3}{10}$
   • d) $\frac{3}{20}$

   Answer: d) $\frac{3}{20}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (3, 6, 9):

   $$HCF(3, 6, 9) = 3$$

   LCM of Denominators (5, 10, 20):

   $$\begin{align*} LCM(5, 10, 20) &= 5 \times 2 \times 2 \\ &= 20 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{3}{20}$$
   Why this question belongs to HCF of Fractions

   This question requires finding the HCF for given fractions, using keywords like 'HCF' (மீ.பொ.வ).

8. $\frac{3}{4}, \frac{6}{7}, \frac{9}{8}$ இவற்றின் HCF காண்க.

   • a) $\frac{3}{28}$
   • b) $\frac{3}{56}$
   • c) $\frac{3}{25}$
   • d) $\frac{3}{32}$

   Answer: b) $\frac{3}{56}$

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (3, 6, 9):

   $$HCF(3, 6, 9) = 3$$

   LCM of Denominators (4, 7, 8):

   $$\begin{align*} LCM(4, 7, 8) &= 4 \times 7 \times 2 \\ &= 56 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{3}{56}$$
   Why this question belongs to HCF of Fractions

   This problem asks for the HCF of fractions, a fundamental concept in this mathematical category.

9. $\frac{2}{3}, \frac{4}{5}, \frac{6}{7}, \frac{8}{9}$ இவற்றின் HCF காண்க.

   • a) $\frac{2}{945}$
   • b) $\frac{2}{385}$
   • c) $\frac{2}{215}$
   • d) $\frac{2}{125}$

   Answer: Correct answer is $\frac{2}{315}$ (Not in options)

   Solution

   பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

   HCF of Numerators (2, 4, 6, 8):

   $$HCF(2, 4, 6, 8) = 2$$

   LCM of Denominators (3, 5, 7, 9): The prime factors are $3=3$, $5=5$, $7=7$, $9=3^2$. The highest powers are $3^2, 5^1, 7^1$.

   $$\begin{align*} LCM(3, 5, 7, 9) &= 3^2 \times 5 \times 7 \\ &= 9 \times 5 \times 7 \\ &= 315 \end{align*}$$

   HCF of Fractions:

   $$HCF = \frac{2}{315}$$

   (Note: The provided source material and options for this question contain a calculation error. The correct LCM is 315, not 945.)

   Why this question belongs to HCF of Fractions

   This question asks to find the HCF (மீ.பொ.வ) of multiple fractions, directly applying the topic's main formula.

10. $\frac{3}{16}, \frac{5}{12}, \frac{7}{8}$ இவற்றின் HCF காண்க.

    • a) $\frac{1}{8}$
    • b) $\frac{1}{16}$
    • c) $\frac{1}{24}$
    • d) $\frac{1}{48}$

    Answer: d) $\frac{1}{48}$

    Solution

    பின்னத்தின் HCF = $\frac{\text{தொகுதிகளின் HCF}}{\text{பகுதிகளின் LCM}}$

    HCF of Numerators (3, 5, 7): Since 3, 5, and 7 are prime numbers, their HCF is 1.

    $$HCF(3, 5, 7) = 1$$

    LCM of Denominators (16, 12, 8):

    $$\begin{align*} LCM(16, 12, 8) &= 2 \times 2 \times 2 \times 2 \times 3 \\ &= 48 \end{align*}$$

    HCF of Fractions:

    $$HCF = \frac{1}{48}$$
    Why this question belongs to HCF of Fractions

    This question involves finding the Highest Common Factor (HCF) for a set of given fractions, a core problem type in this category.