Fractions Decimals and Exact Calculation

What you will learn

A fraction and a decimal can name the same number. Choosing a useful form lets you compare quantities, calculate accurately and recognise when an answer has only been approximated. By the end you should be able to:

• Explain a numerator, denominator and specified whole, and move between mixed and improper fractions
• Make equivalent fractions, simplify them and compare them using common-sized parts
• Add, subtract, multiply and divide fractions, explaining why the methods work
• Use decimal place value and all four decimal operations
• Convert terminating decimals and a simple recurring decimal to exact fractions
• Preserve exact values, round only when requested, and check the sign and size of an answer

This is the fourth lesson in Numbers and Calculation. It develops the fractions and decimals topics named in the RRB Mathematics syllabus. It does not predict question frequency. Percentages, commercial arithmetic and general equation-solving are outside this lesson.

1 Retrieve the tools you already know

Try these before reading the feedback. These are reminders of earlier lessons, not a timed entry test.

1. Find HCF(18, 24).
2. Find LCM(8, 12).
3. Calculate 18 ÷ 3 × 2.
4. Calculate −7 − (−2).

Feedback:

1. 6. It is the greatest positive number dividing both 18 and 24: 18 = 6 × 3 and 24 = 6 × 4. We will use it to simplify a fraction.
2. 24. It is the smallest positive common multiple: 24 = 8 × 3 = 12 × 2. We will use it to choose common-sized parts. Any common multiple would work; the least usually keeps the arithmetic smaller.
3. 12. Division and multiplication share a priority level: 18 ÷ 3 × 2 = 6 × 2 = 12.
4. −5. Subtracting −2 adds its opposite: −7 + 2 = −5.

If a step is uncertain, revisit the sign/order lesson or the HCF/LCM lesson. Here we will retrieve those tools only where needed. There is no score cutoff for continuing.

2 Fractions name equal parts and division

Fix a whole or a unit first. If one strip is divided into five equal pieces, each piece is 1/5 of that strip. Three such pieces are 3/5. The denominator, 5, tells how many equal pieces make one whole; the numerator, 3, counts pieces of that size. Unequal pieces cannot simply be counted as equal fifths.

A fraction also represents division: 3/5 = 3 ÷ 5. Sharing three identical strips equally among five people gives each person 3/5 of a strip. The same notation therefore works for quantities larger than one whole, and for numbers on the number line. To mark 3/5, divide the interval from 0 to 1 into five equal steps and take three steps right from 0.

For positive fractions, a proper fraction has numerator smaller than denominator, such as 3/5. An improper fraction has numerator at least as large as denominator, such as 13/5 or 5/5. A mixed number, such as 2 3/5, records whole units plus a proper fraction: 2 + 3/5. It does not mean 2 × 3/5.

Denominators cannot be zero. A zero numerator is allowed: 0/5 = 0. Keep denominators positive when comparing signed fractions: 3/(−5) = −3/5 and (−3)/(−5) = 3/5. The sign rules from Lesson 1 still apply. A negative fraction lies to the left of zero; for a negative mixed amount, write −(2 + 3/5) so the whole scope of the minus sign is clear.

Equivalent fractions keep the same quantity

Cut each fifth of a strip into two equal pieces. The whole now contains ten pieces and the original three fifths contain six. Thus 3/5 = 6/10. Both the counted pieces and the pieces per whole have doubled; the quantity has not.

Multiplying the numerator and denominator by the same nonzero number preserves a fraction's value. For integer fractions here, use nonzero integer multipliers. Dividing both by the same positive common factor also preserves the value. A fraction is in simplest form when numerator and positive denominator have no common factor greater than 1. HCF gives a one-step route to that form.

Worked example 1 Mixed form and simplification

Express 2 3/5 as an improper fraction, reverse the conversion, and simplify 18/24.

1. Each whole contains five fifths. Two wholes contain 2 × 5 = 10 fifths. With three more fifths, 2 3/5 = (10 + 3)/5 = 13/5.
2. Conversely, 13 ÷ 5 gives quotient 2 and remainder 3: 13 = 2 × 5 + 3. Hence 13/5 = 2 + 3/5.
3. HCF(18, 24) = 6, so 18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4. The HCF of 3 and 4 is 1, so the result is simplest.

Check: 13/5 is between 2 = 10/5 and 3 = 15/5. The reduction of 18/24 must not change its value or its position on the number line.

Error check: Simplification cancels common factors, not matching written digits. For example, 12/15 = 4/5 after dividing by 3; deleting the digit 1 would give 2/5, a different number. Nor can we cancel just one term in (3 + 6)/3: the whole numerator is 9, so the value is 3.

3 Compare fractions using common-sized parts

For a common positive denominator, compare the numerators: 7/12 > 5/12 because seven twelfths exceed five twelfths. With different denominators, first replace both fractions by equivalent ones with a common denominator. When quantities have units, their wholes must also match: half of a large sheet need not be smaller than three quarters of a small sheet.

Worked example 2 Compare two fractions

Which is greater, 5/8 or 7/12?

1. LCM(8, 12) = 24.
2. 5/8 = (5 × 3)/(8 × 3) = 15/24.
3. 7/12 = (7 × 2)/(12 × 2) = 14/24.
4. Since 15 > 14, 5/8 > 7/12. Their difference is one twenty-fourth.

Both fractions are between 1/2 and 1, which is a useful rough check. A common denominator such as 96 would also work; 24 is simply more convenient.

Error check: A larger denominator means smaller individual parts when the whole is fixed, but the number of parts taken also matters. Comparing only 8 and 12 ignores the different numerators. For negatives, the order reverses when both signs are changed: −5/8 = −15/24 < −14/24 = −7/12.

Pause and explain: Why is 4/9 less than 5/9? They count the same-sized ninths, and four is less than five.

4 Add and subtract common units

Two fifths plus one fifth is three fifths: 2/5 + 1/5 = 3/5. The denominator stays 5 because the unit being counted is still a fifth. For different denominators, use equivalent fractions first, then add or subtract the signed numerators. Reduce the result if possible.

Worked example 3 Addition and subtraction together

Calculate 5/6 − 1/4 + 1/3.

1. A common denominator is LCM(6, 4, 3) = 12.
2. 5/6 − 1/4 + 1/3 = 10/12 − 3/12 + 4/12.
3. Work left to right: 10/12 − 3/12 = 7/12; then 7/12 + 4/12 = 11/12.
4. Thus the answer is 11/12, already in simplest form.

Check: −1/4 + 1/3 adds a net 1/12 to 5/6 = 10/12. The answer is positive and just below 1.

Error check: Do not calculate (5 − 1 + 1)/(6 − 4 + 3). Adding or subtracting denominators changes the size of the pieces rather than just their number. Even 1/2 + 1/2 = 1, whereas the wrong rule would give 2/4 = 1/2.

Signs remain meaningful: −2/3 + 1/4 = −8/12 + 3/12 = −5/12. Also 1/4 − (−1/2) = 1/4 + 2/4 = 3/4. Keep a negative fraction in brackets when it follows an operation sign.

5 Multiply as taking a part of a quantity

Multiplying a positive quantity by 3/4 means divide that quantity into four equal parts and take three. When taking a fraction of a fraction, the subdivisions combine. This explains the rule: multiply the numerators and multiply the denominators. A whole number can be written over 1, and a mixed number should first be written as an improper fraction.

Worked example 4 A fraction of a fraction

Find 3/4 of 2/5 of one whole sheet.

1. Imagine the whole sheet split into five equal columns. The first two columns form 2/5 of the sheet.
2. Split every column into four equal pieces. The whole has 5 × 4 = 20 equal pieces. Three of the four pieces in each of the selected two columns give 3 × 2 = 6 pieces.
3. Therefore (3/4) × (2/5) = (3 × 2)/(4 × 5) = 6/20 = 3/10 of the whole sheet.

Check: 3/10 < 2/5 because taking only 3/4 of a positive quantity leaves less than the full quantity. Multiplication need not increase a number. Multiplying a positive quantity by a positive number greater than 1 increases it; multiplying by a number between 0 and 1 decreases it.

You may simplify common factors before multiplying: (3 × 2)/(4 × 5) = (3 × 1)/(2 × 5) = 3/10. This works because 2 divides the entire numerator product and denominator product. It is not cancellation across addition.

Error check: In this sentence, “of” becomes the explicit multiplication (3/4) × (2/5). It does not create a new universal BODMAS priority. Also, −(3/4) × (2/5) = −3/10, by the different-sign rule.

6 Divide by asking what the divisor must multiply

A ÷ B asks for the number which, multiplied by B, gives A; B must be nonzero. A reciprocal is the multiplicative partner that gives 1. For example, (3/8) × (8/3) = 1, so 8/3 is the reciprocal of 3/8. The reciprocal of 4 is 1/4. Zero has no reciprocal.

Why multiply by the reciprocal when dividing? Suppose we need a quantity that gives A after multiplication by 3/8. The quantity A × (8/3) works, because multiplying it by 3/8 gives A × 1 = A. That is why A ÷ (3/8) = A × (8/3), not an instruction to turn over every fraction in sight.

Worked example 5 Equal portions

How many portions of 3/8 litre fit exactly into 9/4 litres?

1. Number of portions = total volume ÷ volume per portion = (9/4) ÷ (3/8).
2. Multiply by the reciprocal of the divisor only: (9/4) × (8/3).
3. Simplify: 9 ÷ 3 = 3 and 8 ÷ 4 = 2, so the answer is 3 × 2 = 6 portions.
4. Check: 6 × (3/8) = 18/8 = 9/4 litres, exactly the starting volume.

A second view uses common units: 9/4 litres = 18/8 litres. Eighteen eighths contain six groups of three eighths. The answer counts portions, so it is 6 portions, not 6 litres.

Error check: Dividing by a positive number below 1 can increase the numerical value. Turning over the dividend instead of the divisor changes the question. If dividing by a mixed number, first convert it to an improper fraction; if the divisor is 0, stop because division is undefined.

For a whole-number divisor, the same method works: (3/5) ÷ 2 = (3/5) × (1/2) = 3/10. For signed values, determine the sign as before: (−3/5) ÷ (−2) = 3/10.

7 Decimals extend place value

The first three places to the right of the decimal point are tenths, hundredths and thousandths. Each place is one tenth of the place immediately to its left. Thus 4.072 = 4 + 0/10 + 7/100 + 2/1000. The zero records that there are no tenths in this place-value expansion.

A finite decimal converts exactly to an integer fraction over 10, 100, 1000 and so on, according to its number of decimal places. Keep the sign, then simplify. Conversely, a fraction is a division: divide its numerator by its denominator, or make the denominator a suitable power of 10 when convenient.

Worked example 6 Comparison and exact conversion

Compare 0.7 and 0.65, then convert 0.375 to a fraction.

1. 0.7 = 7/10 = 70/100 = 0.70. Hence 0.70 > 0.65 because 70 hundredths exceed 65 hundredths.
2. 0.375 has three decimal places, so 0.375 = 375/1000.
3. Divide numerator and denominator by 125: 375/1000 = 3/8.
4. Check by reversing the conversion: 3/8 = (3 × 125)/(8 × 125) = 375/1000 = 0.375.

Error check: More decimal digits do not necessarily mean a larger number. Zeros appended at the end of a finite decimal do not change its value: 0.7 = 0.70 = 0.700. A zero inside the place-value positions can matter: 0.07 is seven hundredths, not seven tenths.

Pause and explain: 0.04 = 4/100 = 1/25, while 0.4 = 4/10 = 2/5. The position of 4 changes its value by a factor of 10.

8 Calculate with decimals by preserving place value

The written-place methods in this section use terminating decimals. For recurring decimals, use an exact fraction; section 9 explains a simple conversion.

For addition and subtraction, line up decimal points so units meet units and tenths meet tenths. Append trailing zeros when helpful. Carrying and regrouping work just as with whole numbers, because ten of any place make one of the place to its left.

For multiplication, convert to fractions to see why the decimal-place counts add. Two decimal places mean a divisor of 100; one more means a divisor of 10; the product has a divisor of 1000. This gives the usual method: multiply the digits as whole numbers, then place the decimal using the total number of decimal places in the factors. Retain any required leading zeros.

For division, multiplying both dividend and nonzero divisor by the same power of 10 preserves the quotient. Choose a power that makes the divisor an integer, then divide, keeping the sign rules. It is the equal scaling of both numbers, not moving a point in just one number, that makes this legal.

Worked example 7 The four decimal operations

Calculate 4.08 + 0.7, 4.08 − 0.7, 0.24 × 0.5 and 3.6 ÷ 0.12.

1. Addition: 4.08 + 0.70 = 408/100 + 70/100 = 478/100 = 4.78. The 7 is in the tenths column, not the hundredths column.
2. Subtraction: 4.08 − 0.70 = 408/100 − 70/100 = 338/100 = 3.38. In column subtraction, regroup 4 units 0 tenths 8 hundredths as 3 units 10 tenths 8 hundredths; removing 7 tenths leaves 3 units 3 tenths 8 hundredths.
3. Multiplication: 0.24 × 0.5 = (24/100) × (5/10) = 120/1000 = 0.120 = 0.12. The two factors have 2 + 1 = 3 decimal places before removing the unnecessary final zero. Half of 0.24 should be 0.12, not 1.2.
4. Division: 3.6 ÷ 0.12 = (3.6 × 100) ÷ (0.12 × 100) = 360 ÷ 12 = 30. Check: 30 × 0.12 = 3.6.

Error check: Aligning the last written digits in addition would treat 0.7 as 0.07. In division, 360 ÷ 0.12 and 3.6 ÷ 12 are both different from the original quotient because only one number was changed.

A quick size check is often enough to reject a misplaced decimal point: a sum of 4.08 and 0.7 should be between 4 and 5; half of a positive number must be smaller; 3.6 contains many portions of size 0.12. With negative decimals, use the same sign rules as for other rational numbers: −0.6 × 0.2 = −0.12. For signed division, 1.5 ÷ (−0.3) = 15 ÷ (−3) = −5: both numbers were multiplied by 10, and the divisor became a negative integer.

Checkpoint: Calculate 0.06 × 0.4 and 1.5 ÷ 0.3 before reading on. 0.06 × 0.4 = 24/1000 = 0.024; 1.5 ÷ 0.3 = 15 ÷ 3 = 5. The first result needs a zero between the decimal point and 2; the second can be checked with 5 × 0.3 = 1.5.

9 Exact decimals and approximations are different

A terminating decimal ends after finitely many decimal places, such as 0.375. A recurring decimal repeats a digit or block forever, possibly after an initial nonrepeating part, such as 0.8333… where only 3 repeats. The dots must refer to a stated continuing pattern. Merely showing a few digits does not prove a pattern.

Both a terminating decimal and a fully specified recurring decimal can give an exact value. A finite rounded version of an infinite recurring decimal is an approximation. Write = for equality and ≈ for an approximation.

Worked example 8 Division explains stopping and repeating

Explain 3/8 = 0.375 and 5/6 = 0.8333…, then express 0.272727… with the block 27 repeating forever as an exact fraction.

For 3 ÷ 8:

1. 3 is less than 8, so write 0 before the decimal point. Express 3 units as 30 tenths. 30 ÷ 8 gives 3 tenths with 6 tenths left.
2. Those 6 tenths are 60 hundredths. 60 ÷ 8 gives 7 hundredths with 4 hundredths left.
3. Those 4 hundredths are 40 thousandths. 40 ÷ 8 gives 5 thousandths with no remainder.
4. Therefore 3/8 = 0.375 exactly. The remainder 0 makes the decimal terminate.

For 5 ÷ 6:

1. 50 tenths divided by 6 gives 8 tenths with 2 tenths left.
2. 20 hundredths divided by 6 gives 3 hundredths with 2 hundredths left.
3. The same remainder calculation repeats in each following place. Thus 5/6 = 0.8333… exactly, with 3 repeating forever. Only the digits after the initial 8 recur.

For 0.272727…:

1. Give the entire recurring number a short name: x = 0.272727…. Here x is just a placeholder for this one number, not a new operation.
2. Since the repeating block has two digits, multiply by 100: 100 × x = 27.272727…. This moves one complete block to the left of the decimal point.
3. Subtract one copy of x from one hundred copies. The identical infinite fractional tails cancel exactly: 100 × x − x = 27, so 99 × x = 27.
4. Divide the 27 equally among 99 copies: x = 27/99 = 3/11.
5. Check by division: 3 ÷ 11 gives 0.27 with remainder 3 after the two decimal places, so the block 27 begins again and continues forever.

The subtraction is valid because both tails represent the same entire recurring fractional value. It is not a subtraction of two decimals that have been cut off after a few digits. We needed 100 because the block contains two digits; a one-digit repeating block uses 10.

Error check: 0.27 = 27/100, while 0.272727… = 3/11. These are not equal. Writing “27 repeats forever” prevents the dots from being read as an unspecified continuation. Do not replace the exact value by 0.27 halfway through a calculation.

A useful link back to prime factors

Every rational number has a terminating or eventually recurring decimal expansion. In long division by a positive integer b, the remainder is one of 0, 1, …, b − 1. Either it becomes 0, or one of the finitely many nonzero remainders repeats. The same remainder starts the same following calculation, so its digits then repeat. After reducing a fraction to simplest form with positive denominator, it has a terminating decimal exactly when that denominator has no prime factors other than 2 and 5; denominator 1 also terminates. Powers of 10 contain only 2s and 5s, so such a denominator can be multiplied up to a power of 10. A remaining different prime factor cannot be removed that way.

For example, 3/8 has denominator 2³ and terminates. The reduced denominator 6 = 2 × 3 in 5/6 still contains 3, so its decimal recurs. Reduce first: 3/6 = 1/2 terminates even though the unreduced denominator 6 contains 3. This test predicts the type of expansion; division determines its digits. A nonterminating, nonrecurring decimal is not rational; that distinction returns with square roots in Lesson 5.

Worked example 9 Round only at the requested step

Round 7.286 to two decimal places.

1. Two decimal places retain tenths and hundredths. The hundredths digit is 8; the next digit, in the thousandths place, is 6.
2. For the usual school rounding rule used here on nonnegative numbers, a next digit from 0 to 4 leaves the retained digit unchanged; 5 to 9 increases it by 1, carrying if necessary.
3. Thus 7.286 ≈ 7.29 to two decimal places. The exact original number remains 7.286 = 7286/1000.
4. Check: 7.286 is 0.004 below 7.29 and 0.006 above 7.28, so 7.29 is the nearer hundredth.

Error check: Simply dropping the final digit gives 7.28; that is truncation, not the requested rounding. Also do not write 7.286 = 7.29. Rounding changes the value here.

Checkpoint: 2.344 ≈ 2.34 and 2.345 ≈ 2.35 to two decimal places under the stated rule. A carry may cross a place: 1.996 ≈ 2.00. The two zeros show the requested two decimal places. Keep exact fractions during intermediate work unless the question explicitly asks for an estimate.

10 Choose an exact form before using BODMAS

Use whichever exact representation makes the work clear. A terminating decimal can become a fraction; a convenient fraction can become a terminating decimal. Then respect brackets, powers, multiplication/division left to right, and addition/subtraction left to right. Check that the evaluated divisor is nonzero.

Worked example 10 Fractions and decimals in one expression

Evaluate (3/4 + 0.5) ÷ (5/8).

1. Convert exactly: 0.5 = 1/2.
2. Evaluate the bracket: 3/4 + 1/2 = 3/4 + 2/4 = 5/4.
3. Divide: (5/4) ÷ (5/8) = (5/4) × (8/5) = 2.
4. Check: the dividend 5/4 = 1.25 is twice the divisor 5/8 = 0.625, and 2 × (5/8) = 5/4.

Error check: Keep the whole sum grouped before dividing. Turning 3/4 + 1/2 into 4/6 would add unlike parts incorrectly. There is no reason to round an exact answer of 2.

A last safety check: (1/2) ÷ (0.75 − 3/4) is undefined, because the grouped divisor is 0. All the written denominators may be nonzero while a calculated divisor still becomes zero.

11 Independent practice

Attempt all twelve questions before viewing the key. Choose one option and record a calculation or reason. The questions are original learning practice, with no time limit, no negative marking and no pass cutoff; they are not a full RRB CBT simulation.

1. What is 21/35 in simplest form?

A. 7/5 B. 3/5 C. 3/7 D. 1/5

2. Write the mixed number 3 2/5 as an improper fraction.

A. 6/5 B. 5/5 C. 11/5 D. 17/5

3. Which comparison is correct?

A. 7/10 > 2/3 B. 7/10 < 2/3 C. 7/10 = 2/3 D. These numbers cannot be compared

4. Which statement correctly compares 0.09 and 0.1?

A. 0.09 > 0.1 B. 0.09 = 0.1 C. 0.09 < 0.1 D. They cannot be compared without rounding

5. Calculate 7/12 + 5/18.

A. 2/5 B. 12/36 C. 31/36 D. 7/6

6. Calculate 5/6 − 3/8.

A. 11/24 B. 1/7 C. −1 D. 2/24

7. Calculate (4/9) × (3/8).

A. 7/17 B. 4/3 C. 1/3 D. 1/6

8. Calculate (5/6) ÷ (10/9).

A. 25/27 B. 3/4 C. 4/3 D. 1/2

9. Calculate 2.75 + 0.6 − 1.08.

A. 2.27 B. 1.73 C. 3.23 D. 2.87

10. A learner claims 0.84 ÷ 0.07 = 1.2. Which correction is valid?

A. 0.12, because only 0.07 becomes 7 B. 120, because the quotient must have one extra zero C. 1.2 is correct because both decimals have two places D. 12, because multiplying both numbers by 100 gives 84 ÷ 7

11. Which fraction equals 0.363636… exactly, when the block 36 repeats forever?

A. 9/25 B. 4/11 C. 36/11 D. 2/11

12. What is the exact value of (2/3 + 1/6) × 1.2?

A. 0.9 B. 1.02 C. 1 D. 1.2

12 Explained answer key

1. B. HCF(21, 35) = 7. Divide both numerator and denominator by 7: 21/35 = 3/5. Since 3 and 5 have no common factor above 1, this is simplest. Dividing only one of the two numbers would change the value.

2. D. 3 2/5 = 3 + 2/5 = 15/5 + 2/5 = 17/5. Check: 17 = 3 × 5 + 2. The mixed number adds two fifths to three wholes; 6/5 comes from incorrectly multiplying 3 by 2/5.

3. A. Use denominator 30: 7/10 = 21/30 and 2/3 = 20/30. Since 21 > 20, 7/10 > 2/3. Different denominators do not prevent comparison; make the parts the same size first.

4. C. 0.1 = 0.10 = 10/100, while 0.09 = 9/100. Nine hundredths are less than ten hundredths. Comparing the digit 9 with the digit 1 while ignoring their place values gives the wrong result; no rounding is needed.

5. C. LCM(12, 18) = 36. Thus 7/12 + 5/18 = 21/36 + 10/36 = 31/36. The numerator 31 shares no factor above 1 with 36. Adding numerators and denominators directly would give 12/30 = 2/5, which does not count common-sized parts.

6. A. LCM(6, 8) = 24. Convert both fractions: 5/6 = 20/24 and 3/8 = 9/24. The difference is (20 − 9)/24 = 11/24. It is positive because 5/6 > 3/8. Subtracting denominators would be invalid.

7. D. (4/9) × (3/8) = 12/72 = 1/6. Or simplify 4 with 8 and 3 with 9 before multiplying, giving (1 × 1)/(3 × 2). The answer is smaller than 4/9 because its positive multiplier 3/8 is below 1. Adding the numerators and denominators is not multiplication.

8. B. Keep the dividend and invert only the nonzero divisor: (5/6) × (9/10) = 45/60 = 3/4. Check: (10/9) × (3/4) = 30/36 = 5/6. The result is below 5/6 because the positive divisor 10/9 is greater than 1.

9. A. Align places and work left to right: 2.75 + 0.60 = 3.35, then 3.35 − 1.08 = 2.27. Equivalently, (275 + 60 − 108)/100 = 227/100. Treating 0.6 as 0.06 would instead produce the incorrect answer 1.73.

10. D. Scale both numbers by 100: 0.84 ÷ 0.07 = 84 ÷ 7 = 12. The check 12 × 0.07 = 0.84 confirms it. The proposed 1.2 fails because 1.2 × 0.07 = 0.084. Only scaling one of the two numbers changes the quotient.

11. B. Let x name the entire recurring decimal. Then 100 × x = 36.363636… . Subtract x: 99 × x = 36, so x = 36/99 = 4/11. The finite decimal 0.36 equals 36/100 = 9/25; it is not the given infinite recurring value.

12. C. First evaluate the bracket: 2/3 + 1/6 = 4/6 + 1/6 = 5/6. Convert 1.2 exactly to 12/10 = 6/5. The product is (5/6) × (6/5) = 1. No rounding is needed. Replacing 2/3 by 0.67 too early would introduce an approximation into an exact calculation.

13 Use errors to choose your next step

• Mixed form or simplification error: return to section 2 and Worked example 1
• Comparison or fraction addition/subtraction error: return to sections 3–4; name the size of each part
• Multiplication/division error: return to sections 5–6; explain the scaling or multiply back
• Decimal place or operation error: return to sections 7–8; use fractions over 10, 100 or 1000 to check
• Recurring-decimal or rounding error: return to section 9; identify the entire repeating block and distinguish = from ≈
• Mixed-expression error: return to section 10; keep brackets and exact values

Cover the key and retry missed items. Also explain why 3/6 terminates after simplification, why 0.27 differs from 0.272727… with 27 repeating forever, and why rounding 2.345 to two decimal places gives 2.35 under our stated rule. If an explanation is unclear, revisit that example rather than chasing a total score. There is no numerical pass gate.

Sources and next step

The explanations, teaching sequence and practice are original. These sources support the mathematical concepts:

• RRB CEN 09/2025, §14.1, printed p.28: syllabus scope includes fractions and decimals; no topic-frequency claim is made
• NIOS Mathematics 211 Chapter 1, §§1.4–1.7, printed pp.11–21: equivalent fractions, comparison and rational arithmetic; the two sections labelled §1.8, pp.22–25: decimal expansions and exact conversions; §1.13, pp.30–31: approximation
• NCERT Ganita Prakash Grade 7 Chapter 3, §3.4, printed p.58 onward; the zero discussion, p.70; §3.7, pp.74–75: decimal place value and addition/subtraction
• NCERT Ganita Prakash Grade 7 Chapter 8, §8.1, printed pp.173–186, and §8.2, pp.186–190: fraction multiplication and division
• NCERT Class VII Fractions and Decimals via IIT Kanpur SATHEE, §§2.1.1, 2.2, 2.3 and 2.4: fraction-of-quantity meaning, reciprocal division and decimal multiplication/division

Next, Squares Square Roots and Sensible Estimates will use fraction simplification, prime factors and exact decimals to explain square roots. No new fraction algorithm is needed before that lesson.

Analogy

Recutting a strip does not change its length

Take one strip as the whole. Mark it into four equal lengths and keep three: you have 3/4 of the strip. Now cut each of those quarter-lengths in half, and imagine the whole marked the same way. The whole has eight equal lengths, and the kept part has six. It is still the same length: 3/4 = 6/8.

This explains why both numerator and denominator change together. Cutting the kept part into more pieces does not create more strip. To add 1/4 of the same whole to 1/8, rewrite 1/4 as 2/8; then 2/8 + 1/8 = 3/8. The counts can be added because each counted piece is now an eighth.

The comparison requires the same whole and equal-sized pieces. It explains equivalence and common-denominator addition; reciprocal division and decimal rounding need the separate explanations in the lesson.

Quick reference

Meaning and equivalent forms

For a/b, b is nonzero. With a positive denominator, it names the part size and the numerator counts those parts. Keep the whole and unit fixed. A mixed number means a sum: 2 3/5 = 2 + 3/5 = 13/5. A fraction also means division.

Multiply numerator and denominator by the same nonzero integer to obtain an equivalent fraction; divide both by a positive common factor to simplify. HCF gives simplest form in one step. Cancel factors of entire products, not digits or isolated terms of sums. Keep denominators positive for comparison.

Fraction operations

• Compare: choose a common positive denominator and compare signed numerators
• Add/subtract: rewrite as common-sized parts, operate on the numerators and keep that denominator
• Multiply: multiply numerators and multiply denominators; common factors may be simplified first
• Divide: multiply the dividend by the reciprocal of the nonzero divisor only; check by multiplying back
• Convert mixed numbers to improper fractions before multiplying or dividing
• For a positive quantity, multiplying by a number between 0 and 1 decreases it; dividing by such a number increases it
• The signed-number rules and BODMAS still apply; a calculated divisor of 0 makes division undefined

Decimal place value and operations

Tenths, hundredths and thousandths correspond to denominators 10, 100 and 1000. Thus 0.375 = 375/1000 = 3/8. Appended trailing zeros preserve value: 0.7 = 0.70, but 0.07 differs.

For terminating decimals, use these written-place methods. For addition/subtraction, align places. For multiplication, the total decimal-place count in the factors gives the placement before unnecessary trailing zeros are removed. For division, scale both dividend and nonzero divisor by the same power of 10 until the divisor is an integer. Check signs and size, then multiply back to check division.

Exact or approximate

A stated infinite repeating pattern gives an exact value: 0.272727… with 27 repeating forever = 3/11, while finite 0.27 = 27/100. Name the entire recurring number x, shift a full block by multiplying by 100, subtract the identical tails, then divide: 99 × x = 27.

Reduce before predicting decimal type. A positive denominator containing no primes except 2 and 5 gives a terminating expansion; denominator 1 also does. Other reduced denominators give recurring expansions.

For the nonnegative numbers rounded here, inspect the first discarded digit: 0–4 keeps the retained digit, 5–9 increases it by 1, carrying if needed. Use ≈ when the value changes: 7.286 ≈ 7.29. Preserve exact values until the requested rounding step. Learning practice has no pass cutoff.