1. HCF and LCM Beyond Whole Numbers
Competitive examinations also ask HCF and LCM of fractions, mixed fractions and decimal numbers. The basic idea remains the same, but the calculation method must first place the numbers in a suitable common form.
Main Cases Covered:
• Proper and improper fractions
• Mixed fractions
• Terminating decimals
• Integers mixed with decimals
• Fractions mixed with terminating decimals
• Reciprocal and common-scaling forms
Exam Convention:
The HCF and LCM formulas for positive fractions used in competitive aptitude are applied after expressing the quantities in exact fractional form, preferably in lowest terms.
2. HCF and LCM of Fractions
For positive fractions, the numerators and denominators are treated differently while calculating HCF and LCM.
HCF of Fractions:
HCF of fractions = HCF of numerators ÷ LCM of denominators
LCM of Fractions:
LCM of fractions = LCM of numerators ÷ HCF of denominators
Memory Shortcut:
HCF of fractions → HCF on top, LCM at bottom.
LCM of fractions → LCM on top, HCF at bottom.
Example — HCF:
Find the HCF of 2/3, 4/9 and 10/21.
HCF of numerators 2, 4, 10 = 2.
LCM of denominators 3, 9, 21 = 63.
Therefore:
HCF = 2/63.
Example — LCM:
Find the LCM of 2/3, 5/6 and 7/9.
LCM of numerators 2, 5, 7 = 70.
HCF of denominators 3, 6, 9 = 3.
Therefore:
LCM = 70/3.
Common Mistake:
Do not use the whole-number rule of taking common factors directly from the fractions without separating numerators and denominators.
3. Simplify Fractions Before Calculation
Fractions should preferably be reduced to their lowest terms before applying HCF-LCM formulas. This makes the calculation smaller and reduces the risk of mistakes.
Example:
Suppose the fractions are 6/8 and 15/20.
Reduce first:
6/8=3/4
15/20=3/4.
Therefore both fractions are equal.
HCF=LCM=3/4.
Exam Tip:
Always look for cancellation before beginning numerator-denominator HCF and LCM calculations.
4. Mixed Fractions
A mixed fraction should first be converted into an improper fraction. After conversion, use the ordinary fraction formulas.
Conversion:
a b/c = (ac+b)/c
Example:
Find the HCF and LCM of 1 1/2 and 2 1/4.
1 1/2=3/2
2 1/4=9/4.
HCF
=HCF(3,9)/LCM(2,4)
=3/4.
LCM
=LCM(3,9)/HCF(2,4)
=9/2.
Therefore:
HCF=3/4
LCM=9/2 = 4 1/2.
Exam Trap:
Do not separately calculate HCF or LCM of the whole-number parts and fractional parts of a mixed fraction.
5. HCF and LCM of Decimal Numbers
For terminating decimals, first make the number of decimal places equal. Then remove the decimal points using the same power of 10 for every number.
Standard Method:
1. Make decimal places equal by adding trailing zeros where necessary.
2. Multiply every number by the same power of 10 to convert them into integers.
3. Find the integer HCF or LCM.
4. Divide the result by the same power of 10.
Example — HCF:
Find the HCF of 1.2, 1.8 and 3.0.
Multiply all by 10:
12, 18, 30.
HCF(12,18,30)=6.
Therefore:
HCF=6/10=0.6.
Example — LCM:
Find the LCM of 0.25, 0.10 and 0.125.
Maximum decimal places=3.
Write them as:
0.250, 0.100, 0.125.
Multiply by 1000:
250, 100, 125.
LCM(250,100,125)=500.
Therefore:
LCM=500/1000=0.5.
Very Important:
All decimal numbers must be multiplied by the same power of 10. Do not remove decimal points independently using different scaling factors.
6. Integers Mixed with Decimals
If some quantities are integers and others are decimals, first write the integers with the required number of decimal places and then apply a common scaling factor.
Example:
Find the LCM of 1.2, 1.8, 2 and 2.5.
Write:
1.2, 1.8, 2.0, 2.5.
Multiply by 10:
12, 18, 20, 25.
LCM(12,18,20,25)=900.
Therefore:
LCM=900/10=90.
Shortcut:
An integer may be treated as a decimal with trailing zeros whenever this helps create a common scale.
7. Fractions and Decimals in the Same Question
When fractions and decimals appear together, convert all quantities into one consistent exact form before calculating HCF or LCM.
Two Useful Approaches:
• Convert all terminating decimals into fractions and use the fraction formulas.
• If all values can easily be represented as terminating decimals with a common scale, convert all of them into scaled integers.
Example:
Find the HCF of 0.5, 3/4 and 1.25.
Convert 3/4=0.75.
Now the values are:
0.50, 0.75, 1.25.
Multiply by 100:
50, 75, 125.
HCF=25.
Therefore:
Required HCF=25/100=0.25.
Exam Tip:
Choose the representation that produces the smallest and cleanest integers.
8. Common Scaling and Reciprocal Shortcuts
The scaling properties learned for integers also remain useful for positive rational numbers.
Common Scaling:
If every number is multiplied by the same positive factor k, then both HCF and LCM are also multiplied by k.
Example:
HCF(12,18,30)=6.
Therefore:
HCF(1.2,1.8,3.0)=0.6.
Reciprocal Property:
For positive rational numbers under the usual aptitude convention:
HCF of reciprocals = reciprocal of the LCM of the original numbers.
LCM of reciprocals = reciprocal of the HCF of the original numbers.
Example:
For 8 and 12:
HCF=4 and LCM=24.
For 1/8 and 1/12:
HCF=1/24
LCM=1/4.
Note:
In most exams, direct fraction formulas are usually safer than relying on reciprocal properties unless the pattern is obvious.
9. Repeating Decimals and Other Exact Forms
If a repeating decimal appears in an exact HCF-LCM question, convert it into an exact fraction first. Do not approximate it.
Example:
0.333... = 1/3.
If such a value appears with fractions, use its exact fractional form before applying HCF-LCM rules.
Exam Trap:
Replacing a repeating decimal by a rounded decimal such as 0.33 changes the exact number and may produce an incorrect HCF or LCM.
10. Choosing the Fastest Method
For ordinary fractions:
Use numerator-denominator formulas directly.
For mixed fractions:
Convert to improper fractions first.
For terminating decimals:
Equalise decimal places and use common scaling.
For fractions mixed with decimals:
Convert everything into whichever exact form gives the cleanest calculation.
For repeating decimals:
Convert to exact fractions first.
11. Common Exam Traps
Trap 1:
Using HCF of denominators while finding HCF of fractions. The denominator requires LCM.
Trap 2:
Using LCM of denominators while finding LCM of fractions. The denominator requires HCF.
Trap 3:
Forgetting to convert mixed fractions into improper fractions.
Trap 4:
Removing decimal points using different powers of 10 for different numbers.
Trap 5:
Forgetting to divide the integer HCF or LCM by the scaling factor after decimal conversion.
Trap 6:
Ignoring trailing zeros needed to make decimal places equal.
Trap 7:
Approximating repeating decimals before an exact HCF-LCM calculation.
12. Quick Revision
Remember:
• HCF of fractions = HCF of numerators / LCM of denominators.
• LCM of fractions = LCM of numerators / HCF of denominators.
• Reduce fractions where possible before calculation.
• Convert mixed fractions into improper fractions first.
• For decimals, equalise decimal places.
• Multiply all decimals by the same power of 10.
• Find integer HCF or LCM and divide by the same scale afterward.
• Integers can be written with trailing decimal zeros.
• Fraction-decimal mixed questions should first be converted into one consistent exact form.
• Repeating decimals should be converted into exact fractions.
• Common scaling changes HCF and LCM by the same scale.
• Always check whether the final result is expressed back in the original scale.
13. Verified Previous-Year Questions
SSC GD PYQ
HCF of Fractions
13 January 2023 · Shift III
Q1. Find the HCF of 3/8, 5/12 and 15/16.
A. 1/12
B. 1/24
C. 1/48
D. 1/36
Correct Answer: C. 1/48
HCF of numerators:
HCF(3,5,15)=1.
LCM of denominators:
LCM(8,12,16)=48.
Therefore:
HCF=1/48.
SSC GD PYQ
HCF of Fractions
6 March 2024 · Shift III
Q2. Find the HCF of 4/5, 3/10 and 7/15.
A. 10/3
B. 1/30
C. 30
D. 1/3
Correct Answer: B. 1/30
HCF of numerators:
HCF(4,3,7)=1.
LCM of denominators:
LCM(5,10,15)=30.
Therefore:
HCF=1/30.
SSC GD PYQ
HCF of Decimals
13 February 2023 · Shift IV
Q3. Find the HCF of 0.25, 2.08 and 8.1.
A. 0.01
B. 0.1
C. 0.05
D. 0.02
Correct Answer: A. 0.01
Write the numbers with two decimal places:
0.25, 2.08, 8.10.
Multiply all by 100:
25, 208, 810.
HCF(25,208,810)=1.
Therefore:
Required HCF=1/100=0.01.
SSC GD PYQ
LCM of Decimals
27 February 2024 · Shift III
Q4. Find the LCM of 1.2, 1.8, 2 and 2.5.
A. 25
B. 30
C. 90
D. 120
Correct Answer: C. 90
Write 2 as 2.0 and multiply all numbers by 10:
12, 18, 20, 25.
LCM(12,18,20,25)=900.
Therefore:
Required LCM=900/10=90.
14. Practice MCQs
Practice MCQ
Q1. Find the HCF of 3/4, 5/6 and 7/8.
A. 1/12
B. 1/18
C. 1/24
D. 1/48
Correct Answer: C. 1/24
HCF(3,5,7)=1 and LCM(4,6,8)=24. Therefore HCF=1/24.
Practice MCQ
Q2. Find the LCM of 2/3, 5/6 and 7/9.
A. 35/3
B. 70/3
C. 70/9
D. 35/9
Correct Answer: B. 70/3
LCM(2,5,7)=70 and HCF(3,6,9)=3. Hence LCM=70/3.
Practice MCQ
Q3. Find the HCF of 1.2, 1.8 and 3.0.
A. 0.2
B. 0.3
C. 0.6
D. 1.2
Correct Answer: C. 0.6
Multiply by 10: 12,18,30. Their HCF is 6. Hence required HCF=0.6.
Practice MCQ
Q4. Find the LCM of 0.4, 0.6 and 1.2.
A. 0.6
B. 1.0
C. 1.2
D. 2.4
Correct Answer: C. 1.2
Multiply by 10: 4,6,12. LCM=12. Therefore required LCM=1.2.
Practice MCQ
Q5. Find the HCF of 1 1/2 and 2 1/4.
A. 1/4
B. 1/2
C. 3/4
D. 3/2
Correct Answer: C. 3/4
The fractions are 3/2 and 9/4. HCF=HCF(3,9)/LCM(2,4)=3/4.
Practice MCQ
Q6. Find the LCM of 1 1/2 and 2 1/4.
A. 3
B. 7/2
C. 4
D. 9/2
Correct Answer: D. 9/2
Using 3/2 and 9/4: LCM=LCM(3,9)/HCF(2,4)=9/2=4 1/2.
Practice MCQ
Q7. Find the HCF of 0.5, 3/4 and 1.25.
A. 0.10
B. 0.20
C. 0.25
D. 0.50
Correct Answer: C. 0.25
3/4=0.75. Scale 0.50,0.75,1.25 by 100 to get 50,75,125. Their HCF is 25. Therefore answer=0.25.
Practice MCQ
Q8. Find the LCM of 0.25, 0.5 and 1.5.
A. 0.5
B. 1
C. 1.5
D. 3
Correct Answer: C. 1.5
Scale by100: 25,50,150. Their LCM is150. Therefore required LCM=1.5.
Practice MCQ
Q9. Find the LCM of 1/8 and 1/12.
A. 1/48
B. 1/24
C. 1/8
D. 1/4
Correct Answer: D. 1/4
LCM of numerators=1 and HCF of denominators HCF(8,12)=4. Hence LCM=1/4.
Practice MCQ
Q10. Find the HCF of 2.4, 3.6 and 6.0.
A. 0.6
B. 1.2
C. 1.8
D. 2.4
Correct Answer: B. 1.2
Scale by10: 24,36,60. Their HCF is12. Therefore required HCF=1.2.
1. HCF and LCM Beyond Whole Numbers
Competitive examinations also ask HCF and LCM of fractions, mixed fractions and decimal numbers. The basic idea remains the same, but the calculation method must first place the numbers in a suitable common form.
Main Cases Covered:
• Proper and improper fractions
• Mixed fractions
• Terminating decimals
• Integers mixed with decimals
• Fractions mixed with terminating decimals
• Reciprocal and common-scaling forms
Exam Convention:
The HCF and LCM formulas for positive fractions used in competitive aptitude are applied after expressing the quantities in exact fractional form, preferably in lowest terms.
2. HCF and LCM of Fractions
For positive fractions, the numerators and denominators are treated differently while calculating HCF and LCM.
HCF of Fractions:
HCF of fractions = HCF of numerators ÷ LCM of denominators
LCM of Fractions:
LCM of fractions = LCM of numerators ÷ HCF of denominators
Memory Shortcut:
HCF of fractions → HCF on top, LCM at bottom.
LCM of fractions → LCM on top, HCF at bottom.
Example — HCF:
Find the HCF of 2/3, 4/9 and 10/21.
HCF of numerators 2, 4, 10 = 2.
LCM of denominators 3, 9, 21 = 63.
Therefore:
HCF = 2/63.
Example — LCM:
Find the LCM of 2/3, 5/6 and 7/9.
LCM of numerators 2, 5, 7 = 70.
HCF of denominators 3, 6, 9 = 3.
Therefore:
LCM = 70/3.
Common Mistake:
Do not use the whole-number rule of taking common factors directly from the fractions without separating numerators and denominators.
3. Simplify Fractions Before Calculation
Fractions should preferably be reduced to their lowest terms before applying HCF-LCM formulas. This makes the calculation smaller and reduces the risk of mistakes.
Example:
Suppose the fractions are 6/8 and 15/20.
Reduce first:
6/8=3/4
15/20=3/4.
Therefore both fractions are equal.
HCF=LCM=3/4.
Exam Tip:
Always look for cancellation before beginning numerator-denominator HCF and LCM calculations.
4. Mixed Fractions
A mixed fraction should first be converted into an improper fraction. After conversion, use the ordinary fraction formulas.
Conversion:
a b/c = (ac+b)/c
Example:
Find the HCF and LCM of 1 1/2 and 2 1/4.
1 1/2=3/2
2 1/4=9/4.
HCF
=HCF(3,9)/LCM(2,4)
=3/4.
LCM
=LCM(3,9)/HCF(2,4)
=9/2.
Therefore:
HCF=3/4
LCM=9/2 = 4 1/2.
Exam Trap:
Do not separately calculate HCF or LCM of the whole-number parts and fractional parts of a mixed fraction.
5. HCF and LCM of Decimal Numbers
For terminating decimals, first make the number of decimal places equal. Then remove the decimal points using the same power of 10 for every number.
Standard Method:
1. Make decimal places equal by adding trailing zeros where necessary.
2. Multiply every number by the same power of 10 to convert them into integers.
3. Find the integer HCF or LCM.
4. Divide the result by the same power of 10.
Example — HCF:
Find the HCF of 1.2, 1.8 and 3.0.
Multiply all by 10:
12, 18, 30.
HCF(12,18,30)=6.
Therefore:
HCF=6/10=0.6.
Example — LCM:
Find the LCM of 0.25, 0.10 and 0.125.
Maximum decimal places=3.
Write them as:
0.250, 0.100, 0.125.
Multiply by 1000:
250, 100, 125.
LCM(250,100,125)=500.
Therefore:
LCM=500/1000=0.5.
Very Important:
All decimal numbers must be multiplied by the same power of 10. Do not remove decimal points independently using different scaling factors.
6. Integers Mixed with Decimals
If some quantities are integers and others are decimals, first write the integers with the required number of decimal places and then apply a common scaling factor.
Example:
Find the LCM of 1.2, 1.8, 2 and 2.5.
Write:
1.2, 1.8, 2.0, 2.5.
Multiply by 10:
12, 18, 20, 25.
LCM(12,18,20,25)=900.
Therefore:
LCM=900/10=90.
Shortcut:
An integer may be treated as a decimal with trailing zeros whenever this helps create a common scale.
7. Fractions and Decimals in the Same Question
When fractions and decimals appear together, convert all quantities into one consistent exact form before calculating HCF or LCM.
Two Useful Approaches:
• Convert all terminating decimals into fractions and use the fraction formulas.
• If all values can easily be represented as terminating decimals with a common scale, convert all of them into scaled integers.
Example:
Find the HCF of 0.5, 3/4 and 1.25.
Convert 3/4=0.75.
Now the values are:
0.50, 0.75, 1.25.
Multiply by 100:
50, 75, 125.
HCF=25.
Therefore:
Required HCF=25/100=0.25.
Exam Tip:
Choose the representation that produces the smallest and cleanest integers.
8. Common Scaling and Reciprocal Shortcuts
The scaling properties learned for integers also remain useful for positive rational numbers.
Common Scaling:
If every number is multiplied by the same positive factor k, then both HCF and LCM are also multiplied by k.
Example:
HCF(12,18,30)=6.
Therefore:
HCF(1.2,1.8,3.0)=0.6.
Reciprocal Property:
For positive rational numbers under the usual aptitude convention:
HCF of reciprocals = reciprocal of the LCM of the original numbers.
LCM of reciprocals = reciprocal of the HCF of the original numbers.
Example:
For 8 and 12:
HCF=4 and LCM=24.
For 1/8 and 1/12:
HCF=1/24
LCM=1/4.
Note:
In most exams, direct fraction formulas are usually safer than relying on reciprocal properties unless the pattern is obvious.
9. Repeating Decimals and Other Exact Forms
If a repeating decimal appears in an exact HCF-LCM question, convert it into an exact fraction first. Do not approximate it.
Example:
0.333... = 1/3.
If such a value appears with fractions, use its exact fractional form before applying HCF-LCM rules.
Exam Trap:
Replacing a repeating decimal by a rounded decimal such as 0.33 changes the exact number and may produce an incorrect HCF or LCM.
10. Choosing the Fastest Method
For ordinary fractions:
Use numerator-denominator formulas directly.
For mixed fractions:
Convert to improper fractions first.
For terminating decimals:
Equalise decimal places and use common scaling.
For fractions mixed with decimals:
Convert everything into whichever exact form gives the cleanest calculation.
For repeating decimals:
Convert to exact fractions first.
11. Common Exam Traps
Trap 1:
Using HCF of denominators while finding HCF of fractions. The denominator requires LCM.
Trap 2:
Using LCM of denominators while finding LCM of fractions. The denominator requires HCF.
Trap 3:
Forgetting to convert mixed fractions into improper fractions.
Trap 4:
Removing decimal points using different powers of 10 for different numbers.
Trap 5:
Forgetting to divide the integer HCF or LCM by the scaling factor after decimal conversion.
Trap 6:
Ignoring trailing zeros needed to make decimal places equal.
Trap 7:
Approximating repeating decimals before an exact HCF-LCM calculation.
12. Quick Revision
Remember:
• HCF of fractions = HCF of numerators / LCM of denominators.
• LCM of fractions = LCM of numerators / HCF of denominators.
• Reduce fractions where possible before calculation.
• Convert mixed fractions into improper fractions first.
• For decimals, equalise decimal places.
• Multiply all decimals by the same power of 10.
• Find integer HCF or LCM and divide by the same scale afterward.
• Integers can be written with trailing decimal zeros.
• Fraction-decimal mixed questions should first be converted into one consistent exact form.
• Repeating decimals should be converted into exact fractions.
• Common scaling changes HCF and LCM by the same scale.
• Always check whether the final result is expressed back in the original scale.
13. Verified Previous-Year Questions
SSC GD PYQ
HCF of Fractions
13 January 2023 · Shift III
Q1. Find the HCF of 3/8, 5/12 and 15/16.
A. 1/12
B. 1/24
C. 1/48
D. 1/36
Correct Answer: C. 1/48
HCF of numerators:
HCF(3,5,15)=1.
LCM of denominators:
LCM(8,12,16)=48.
Therefore:
HCF=1/48.
SSC GD PYQ
HCF of Fractions
6 March 2024 · Shift III
Q2. Find the HCF of 4/5, 3/10 and 7/15.
A. 10/3
B. 1/30
C. 30
D. 1/3
Correct Answer: B. 1/30
HCF of numerators:
HCF(4,3,7)=1.
LCM of denominators:
LCM(5,10,15)=30.
Therefore:
HCF=1/30.
SSC GD PYQ
HCF of Decimals
13 February 2023 · Shift IV
Q3. Find the HCF of 0.25, 2.08 and 8.1.
A. 0.01
B. 0.1
C. 0.05
D. 0.02
Correct Answer: A. 0.01
Write the numbers with two decimal places:
0.25, 2.08, 8.10.
Multiply all by 100:
25, 208, 810.
HCF(25,208,810)=1.
Therefore:
Required HCF=1/100=0.01.
SSC GD PYQ
LCM of Decimals
27 February 2024 · Shift III
Q4. Find the LCM of 1.2, 1.8, 2 and 2.5.
A. 25
B. 30
C. 90
D. 120
Correct Answer: C. 90
Write 2 as 2.0 and multiply all numbers by 10:
12, 18, 20, 25.
LCM(12,18,20,25)=900.
Therefore:
Required LCM=900/10=90.
14. Practice MCQs
Practice MCQ
Q1. Find the HCF of 3/4, 5/6 and 7/8.
A. 1/12
B. 1/18
C. 1/24
D. 1/48
Correct Answer: C. 1/24
HCF(3,5,7)=1 and LCM(4,6,8)=24. Therefore HCF=1/24.
Practice MCQ
Q2. Find the LCM of 2/3, 5/6 and 7/9.
A. 35/3
B. 70/3
C. 70/9
D. 35/9
Correct Answer: B. 70/3
LCM(2,5,7)=70 and HCF(3,6,9)=3. Hence LCM=70/3.
Practice MCQ
Q3. Find the HCF of 1.2, 1.8 and 3.0.
A. 0.2
B. 0.3
C. 0.6
D. 1.2
Correct Answer: C. 0.6
Multiply by 10: 12,18,30. Their HCF is 6. Hence required HCF=0.6.
Practice MCQ
Q4. Find the LCM of 0.4, 0.6 and 1.2.
A. 0.6
B. 1.0
C. 1.2
D. 2.4
Correct Answer: C. 1.2
Multiply by 10: 4,6,12. LCM=12. Therefore required LCM=1.2.
Practice MCQ
Q5. Find the HCF of 1 1/2 and 2 1/4.
A. 1/4
B. 1/2
C. 3/4
D. 3/2
Correct Answer: C. 3/4
The fractions are 3/2 and 9/4. HCF=HCF(3,9)/LCM(2,4)=3/4.
Practice MCQ
Q6. Find the LCM of 1 1/2 and 2 1/4.
A. 3
B. 7/2
C. 4
D. 9/2
Correct Answer: D. 9/2
Using 3/2 and 9/4: LCM=LCM(3,9)/HCF(2,4)=9/2=4 1/2.
Practice MCQ
Q7. Find the HCF of 0.5, 3/4 and 1.25.
A. 0.10
B. 0.20
C. 0.25
D. 0.50
Correct Answer: C. 0.25
3/4=0.75. Scale 0.50,0.75,1.25 by 100 to get 50,75,125. Their HCF is 25. Therefore answer=0.25.
Practice MCQ
Q8. Find the LCM of 0.25, 0.5 and 1.5.
A. 0.5
B. 1
C. 1.5
D. 3
Correct Answer: C. 1.5
Scale by100: 25,50,150. Their LCM is150. Therefore required LCM=1.5.
Practice MCQ
Q9. Find the LCM of 1/8 and 1/12.
A. 1/48
B. 1/24
C. 1/8
D. 1/4
Correct Answer: D. 1/4
LCM of numerators=1 and HCF of denominators HCF(8,12)=4. Hence LCM=1/4.
Practice MCQ
Q10. Find the HCF of 2.4, 3.6 and 6.0.
A. 0.6
B. 1.2
C. 1.8
D. 2.4
Correct Answer: B. 1.2
Scale by10: 24,36,60. Their HCF is12. Therefore required HCF=1.2.