Converting a terminating or repeating decimal back to p/q

What to assume they know

• Why long division must either stop or loop — that a fraction's decimal stops or repeats, and what a repeating block is
• Bar notation for a repeating block
• Multiplying a decimal by a power of ten, and what that does to the point
• Solving a one-step linear equation
• Reducing a fraction to lowest terms
• Reading a terminating decimal as a fraction over a power of ten

What they should be able to do

• Convert a terminating decimal to a fraction and reduce it
• Identify, in a repeating decimal, which digits repeat and which do not, and count each group
• Convert a pure repeating decimal to a fraction by one multiplication and a subtraction
• Convert a general repeating decimal by two multiplications and a subtraction, and say what each multiplication is for
• Explain why the subtraction removes the infinite tail exactly
• Choose the correct powers of ten from the counts of repeating and non-repeating digits
• Apply the method to show that a decimal of all nines equals a whole number
• State the two rows of the chapter's conversion table from memory
• Recognise that a terminating decimal has a second writing ending in repeated nines

Where it usually goes wrong

• "Multiply by 10 always." The power of ten is set by the block length. A two-digit block needs 100, and using 10 leaves a tail behind that will not subtract away.
• "Subtracting infinite decimals is approximate." It is exact here, and only here, because the two tails were engineered to be identical digit for digit. Subtracting two different infinite tails would not be legitimate, and saying so is what keeps the method honest.
• "For the general case you multiply by one big power of ten." You need two separate shifts with different purposes: the first parks the non-repeating digits to the left of the point, the second advances by exactly one cycle. Collapsing them into one step is where students lose the method.
• "0.999… is just under 1." The chapter says outright that many people expect this, and the algebra in Q4 shows it is wrong. This is the item most likely to provoke argument in a classroom.
• "Every number has one decimal writing." Terminating decimals have two. The chapter's own non-uniqueness paragraph exists to say so, and it is the reason the decimal expansion is a signature but not a fingerprint.
• "2.47 and 2.46999… differ in the third decimal place." They are the same number written two ways, which is exactly what the paragraph on p. 62 asserts.
• "An overline over the last digit means the whole decimal repeats." In end-of-chapter Q3 the bars sit over different-sized groups in different items, and five of the nine have digits outside the bar. Misreading a bar's extent produces the wrong power of ten and then the wrong fraction.
• "You are finished when you have solved for x." You are finished when the fraction is reduced. All five printed examples reduce at the last step — 6/9 to 2/3, 45/99 to 5/11, 15/90 to 1/6, 2122/900 to 1061/450 and 24292/9900 to 6073/2475 — so reduction is the rule here, not the exception.

Questions to check understanding

• Convert a terminating decimal to a fraction in lowest terms
• Convert a pure repeating decimal to a fraction, showing the multiplication and subtraction
• Convert a general repeating decimal, stating both powers of ten and why each was chosen
• Show algebraically that a decimal of all nines equals 1
• Given a bar over a stated group of digits, identify the counts of repeating and non-repeating digits
• The chapter's own nine-item conversion list, in whole or in part
• Explain why a terminating decimal has a second writing
• Solve a linear equation in a rational unknown, and reason about the sign of a product without numerical values

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated inputs. All the printed examples below give their own results.

• Example 4, the terminating case (§3.6.1, p. 59). 0.35 is read as 35/100 and reduced to 7/20. Verified. The chapter notes that this direction was covered in earlier classes and spends no more time on it.
• Example 5, a one-digit block (p. 59). Inputs: x set equal to a decimal of repeating 6s. Steps as printed: one digit repeats, so multiply by 10¹, giving 10x as 6 point 6 recurring; subtract the first equation from the second, giving 6 on the right; so 9x = 6 and x = 6/9 = 2/3. Verified.
• Example 6, a two-digit block (p. 59). Inputs: x set equal to a decimal of repeating 45s. Two digits repeat, so multiply by 10² = 100, giving 100x as 45 point 45 recurring; subtracting gives 99x = 45, so x = 45/99 = 5/11. Verified. Note the tie-back: Example 3 on p. 57 produced this same decimal from 5/11 by division, so the two directions meet on one number. Say so — it is the cheapest demonstration that the method is an inverse.
• Example 7, one non-repeating digit and one repeating (pp. 59–60). Inputs: x set equal to 0.1 followed by repeating 6s. Steps as printed: one digit does not repeat, so multiply by 10¹ to get 10x as 1 point 6 recurring; then multiply by 10¹ again to move one full cycle, giving 100x as 16 point 6 recurring; taking the earlier shifted equation away from the later one gives 90x = 15, so x = 15/90 = 1/6. Verified.
• Example 8, two non-repeating digits and one repeating (p. 60). Inputs: x set equal to 2.35 followed by repeating 7s. Printed steps: multiply by 10² = 100 to clear the two non-repeating digits, giving 100x as 235 point 7 recurring; multiply by 10¹ to move the single-digit cycle, giving 1000x as 2357 point 7 recurring; subtract to get 900x = 2122, so x = 2122/900 = 1061/450. Verified.
• Example 9, two non-repeating and two repeating (p. 60). Inputs: x set equal to 2.45 followed by a repeating 37. Printed steps: multiply by 10² = 100 giving 100x as 245 point 37 recurring; multiply by 10² again giving 10000x as 24537 point 37 recurring; subtract to get 9900x = 24292, so x = 24292/9900 = 6073/2475. Verified; the common factor removed is 4.
• The pattern in the multipliers (an added observation, from the chapter's five examples). Verified across all five: the final coefficient of x is 10^(m+n) − 10^m, where m counts the non-repeating digits and n the repeating ones. For Example 5 that is 10 − 1 = 9; Example 6, 100 − 1 = 99; Example 7, 100 − 10 = 90; Example 8, 1000 − 100 = 900; Example 9, 10000 − 100 = 9900. Putting the five coefficients in a column makes the structure visible in a way the worked examples individually do not.
• The chapter's Summary Table for Conversion (pp. 60–61). Two rows. Pure repeating: set x to the decimal, multiply by ten raised to the count of repeating digits, subtract from the original equation, solve. General repeating: set x to the decimal, multiply by ten raised to the count of non-repeating digits, then by ten raised to the count of repeating digits, subtract from the previous equation, solve. The table straddles the page turn.
• Nine recurring (Exercise Set 3.5, Q4, p. 62). Inputs as printed: the decimal of repeating 9s is stated to be a rational number, and the student is directed to set x equal to it, multiply by 10, subtract, and explain why the value is exactly 1. Verified: 10x − x = 9, so 9x = 9 and x = 1. This is the same Example 5 machinery on a different digit, which is why it belongs here rather than in the irrationals topic.
• The non-uniqueness paragraph (p. 62, under a bold run-in heading). Printed inputs: 1 written as 10/10 and as 100/100; the statement that any terminating decimal has an alternative form with repeating 9s; and the two instances 1.000… equal to 0.999… and 2.47000… equal to 2.46999…. The chapter then remarks that many would have expected the nines form to fall short of 1. Read from p. 62.
• End-of-chapter Q1 (p. 64). Convert by long division: 3/50 and 2/9. Verified: 0.06, terminating; and 0.222… with a one-digit block. This is the opposite direction and makes a good bookend.
• End-of-chapter Q3 (p. 64), the largest exercise in the chapter, read from the printed page because the overlines do not extract. Nine items, with the bars as printed: (i) 12.6 with no bar; (ii) 0.0120 with no bar; (iii) 3.0 followed by a repeating 52; (iv) 1.2 followed by a repeating 35; (v) a repeating 23 straight after the point; (vi) 2.0 followed by a repeating 5; (vii) 2.12 followed by a repeating 5; (viii) 3.12 followed by a repeating 5; (ix) 2 followed by a point and then a repeating 1625 — mind the integer part here, since dropping it turns 2.1625… into 0.1625… and yields the wrong fraction. So items (i) and (ii) are terminating, (v) is the one pure repeating case, and the rest, (ix) included, are general repeating. Verified for two of them: (i) 12.6 = 63/5; (v) the repeating 23 gives 23/99. Hand the rest over as inputs.
• End-of-chapter Q8 and Q9 (p. 65), adjacent items that use fractions rather than decimals: Q8 asks for the rational x with x/3 + x/5 = 16/15; Q9 asks, for non-zero rationals a and b with a + 1/b = 0, whether ab is positive or negative, with a justification and no numerical values assigned. Verified: Q8 gives x = 2; Q9 gives ab negative, since a = −1/b makes ab = −1.

Figures to have open

• A tail-alignment graphic: the original decimal and its shifted copy stacked so that identical digits sit in identical columns, with the cancelling region shaded. This is the figure the whole topic depends on and the chapter prints nothing like it.
• The chapter's two-row conversion table re-set as a clean reference card (pp. 60–61). Re-set it rather than reproducing the printed panel; note that it straddles a page turn in the book, so a single card is an improvement.
• A counts-to-powers strip: the number of non-repeating digits and the number of repeating digits feeding into the two multipliers. Standard schematic.
• A column of the five printed examples' coefficients for section 9. Built from the values above.
• No textbook figure is needed; pp. 59 and 60 are worked equations set as text, which I confirmed on the printed pages.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 3 on the world of numbers. Within §3.6.1 the chapter prints an unnumbered bold subheading on converting rational decimals into fractional form, beginning on p. 58; Case 1 sits on p. 58, Example 4 and Cases 2 and 3 on p. 59, Examples 8 and 9 on p. 60.
• The Summary Table for Conversion begins on p. 60 and its second row completes on p. 61.
• Exercise Set 3.5, Q4, p. 62, and the bold run-in paragraph on non-uniqueness at the foot of the same page.
• End-of-chapter exercises Q1 and Q3, p. 64, and Q8 and Q9, p. 65.
• Backward pointer: the decimal in Example 6 is the same number produced by division in Example 3, §3.6.1, p. 57.