Fractions, Decimals, LCM and HCF
Fractions, Decimals, LCM and HCF
This topic supplies the most predictable question in the whole maths block. Expect 1 to 2 questions, usually a straight LCM or HCF word problem about bells, tiles or bundles. The arithmetic is class 6-8, so the marks are decided by speed and by reading the question correctly.
Comparing and Converting Fractions
To compare two fractions, cross multiply and compare the products.
a/b vs c/d → compare (a × d) with (b × c)
For 3/7 and 4/9: 3 × 9 = 27 and 7 × 4 = 28. Since 27 < 28, 3/7 is smaller.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333 | 33.33% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 1/16 | 0.0625 | 6.25% |
| 2/3 | 0.666 | 66.67% |
Memorising this table is the single highest-return thing you can do for the arithmetic block. It is reused in percentage, profit-loss and data interpretation.
HCF and LCM: What Each One Actually Means
HCF is the largest number that divides all the given numbers. LCM is the smallest number that all the given numbers divide into. Read the word problem and ask which of the two the situation needs.
- Cutting something into the largest equal pieces → HCF
- Events repeating together after some time → LCM
- Largest container that measures several quantities exactly → HCF
- Smallest number leaving the same remainder with several divisors → LCM
HCF × LCM = Product of the two numbers
This identity works only for two numbers. If HCF of two numbers is 6 and their LCM is 36, and one number is 12, the other is (6 × 36) ÷ 12 = 18.
✗ HCF of 12 and 18 is 36 | ✓ HCF of 12 and 18 is 6; 36 is their LCM
A Worked Word Problem
Three school bells in a Muzaffarpur middle school ring at intervals of 12, 15 and 20 minutes. If they ring together at 9:00 am, when do they next ring together?
Take the LCM of 12, 15 and 20. LCM = 60. So they ring together again at 10:00 am. Notice the wording — ring together again is your signal for LCM.
साथ में होना = ल.स., बराबर बांटना = म.स. — यही एक लाइन पूरा सवाल हल कर देती है।
TRE pointer: LCM-HCF word problems are among the safest marks in Part-II because the numbers stay small and the method never changes. The trap is a question that gives you three numbers and asks for the greatest number that divides them leaving remainders 2, 3 and 4 — here you first subtract the remainders, then take the HCF. Bihar-flavoured versions often use school bells, Kosi embankment lamp posts or fair-price shop bundles as the wrapper.
60-Second Recap
- Cross multiply to compare fractions; memorise the fraction-decimal-percent table.
- HCF = biggest common divisor. LCM = smallest common multiple.
- Largest equal pieces → HCF. Events repeating together → LCM.
- HCF × LCM = product of the two numbers (two numbers only).
- With remainders given, subtract them first and then take HCF.
- Expect 1-2 questions from this topic in the Part-II maths block.
Frequently Asked Questions
When should I use LCM and when HCF in a word problem?
Ask what the answer looks like. If the answer is a big time or a big number after which things repeat together, use LCM. If the answer is the largest size, largest container or maximum equal pieces, use HCF. The words together and again point to LCM; the words largest and maximum point to HCF.
Does HCF times LCM equal the product for three numbers?
No. The identity HCF × LCM = product of numbers holds only for two numbers. For three or more numbers it fails, and questions sometimes test exactly this. If a question gives three numbers with their HCF and LCM, solve it by prime factorisation instead of using the shortcut.
What is the fastest way to find LCM of three numbers?
Use the division method. Write the three numbers in a row, divide repeatedly by the smallest prime that divides at least one of them, carrying down the numbers unchanged when they do not divide, and multiply all the divisors at the end. For small class 6-8 numbers this takes under twenty seconds.
Are calculators allowed in BPSC TRE 4.0?
No. All calculation in both the Prarambhik and Mukhya papers has to be done by hand, which is why the fraction-decimal table and divisibility rules matter so much. The numbers are kept small deliberately, so accuracy under time pressure is what the paper is really testing.