Chapter 1.2 — Exercise 1.1 — Rational Numbers Revision

Squeezing Rational Numbers Between Two Fractions

Questions 3 and 4 apply the common-denominator method directly. For five rationals between 1 and 2, use denominator 6 (one more than the count needed): 1 = 6/6 and 2 = 12/6, leaving 7/6, 8/6, 9/6, 10/6, and 11/6 in between.

Question 4 starts with fractions instead of whole numbers, which adds a step: insert three rationals between 3/5 and 2/3. First equalise the denominators — 3/5 = 9/15 and 2/3 = 10/15 — then, since 9/15 and 10/15 are adjacent with no room between them yet, scale both by 4 to open up space: 36/60 and 40/60. Now 37/60, 38/60, and 39/60 sit cleanly in between.

The scaling step is the part most easily rushed. 9/15 and 10/15 are already adjacent — there's no whole number sitting between 9 and 10 — so multiplying by 4 isn't optional, it's what opens up the three extra slots (36, 37, 38, 39, 40) needed to fit three brand-new numbers strictly between the original two. Multiplying by 2 would only have opened one slot; multiplying by 5 would have opened four. The multiplier has to be at least one more than the count of numbers required.

Marking Fifths on the Number Line

Question 5 asks for 8/5 and −8/5 to be plotted. Converting to a mixed number first makes the placement obvious: 8/5 = 1⅗, so it lies between 1 and 2, specifically at the third of five equal divisions past 1. The negative version, −8/5 = −1⅗, mirrors this on the other side of zero, landing between −1 and −2. The number of equal parts the unit interval gets divided into always matches the denominator — five parts, because the denominator is 5.

It's worth noticing that 8/5 and −8/5 are mirror images of each other across zero — same distance, opposite direction. That symmetry holds for every rational number and its negative, which is a quick way to sanity-check a number-line placement: if the positive version sits a third of the way past 1, the negative version must sit exactly a third of the way past −1, not somewhere that merely looks close.

Eight Fractions, Two Decimal Behaviours

Question 6 hands over eight fractions and asks for their decimal form by direct division. The first four terminate cleanly; the last four settle into a repeating block instead.

• 242/1000 = 0.242 (terminating)
• 354/500 = 0.708 (terminating)
• 2/5 = 0.4 (terminating)
• 115/4 = 28.75 (terminating)
• 2/3 = 0.6̄ (recurring)
• −25/36 = −0.69̄4̄ (recurring — only the final 4 repeats)
• 22/7 = 3.1̄42857̄ (recurring — the familiar π approximation, itself rational)
• 11/9 = 1.2̄ (recurring)

A pattern worth noticing across all eight: every terminating case has a denominator built only from 2s and 5s once written in lowest terms (1000, 500, 5, and 4 all qualify), while every recurring case has some other prime lurking in its denominator (3 for 2/3, 36 = 2²×3² for −25/36, 7 for 22/7, 9 = 3² for 11/9). That's the same rule from the chapter introduction, now confirmed across eight fresh examples rather than just the two it was first shown with.

Turning Decimals Back Into Fractions

Question 7 covers terminating decimals: place the digits over the matching power of 10 and simplify. 10.25 = 1025/100 = 41/4; 15.4 = 154/10 = 77/5. The power of 10 used in the denominator always matches the number of digits after the decimal point — two digits after the point means a denominator of 100, one digit means 10 — and simplifying afterward is what turns an ugly fraction like 1025/100 into the much cleaner 41/4.

Question 8 covers recurring decimals, which need one of three sub-rules depending on the pattern:

• Purely recurring (e.g. 0.5̄): place the repeating block over as many 9s as there are repeating digits → 0.5̄ = 5/9.
• Two-digit recurring block (e.g. 0.3̄6̄): place over 99 → 0.3̄6̄ = 36/99 = 4/11.
• Mixed, part recurring (e.g. 3.12̄7̄): subtract the non-recurring portion from the whole thing before dividing → 3.12̄7̄ = 3 + (127 − 12)/900 = 3 + 115/900 = 563/180.

The denominator in the mixed case deserves a closer look, since it's the step most often done by rote. 900 comes from two 9s (matching the two repeating digits, "27") followed by one 0 (matching the one non-repeating digit after the decimal point, "1"). Change the count of repeating or non-repeating digits and the number of 9s and 0s changes to match — the pattern always tracks the digit count exactly, never a fixed value.

Reading a Denominator Before You Divide

Question 9 puts the terminating-decimal shortcut from the chapter introduction to direct use — check only the prime factors of the denominator, no long division required.

• 3/25: 25 = 5² = 2⁰ × 5² → terminating
• 11/18: 18 = 2 × 3² → factor of 3 present → recurring
• 13/20: 20 = 2² × 5 → terminating
• 41/42: 42 = 2 × 3 × 7 → factors of 3 and 7 present → recurring

This shortcut only works reliably once the fraction is reduced to lowest terms first. Checked carelessly, 3/6 looks like it should recur, since its denominator 6 = 2 × 3 carries a factor of 3. But 3/6 reduces to 1/2 before the rule is meant to apply, and 1/2 = 0.5 terminates without any trouble. The rule is always checking the denominator after cancelling common factors with the numerator, never the denominator as first written.

Carrying These Skills Into Irrational Numbers

Every technique in this exercise — locating fractions precisely, inserting values between two numbers, converting decimals in both directions — assumes the number in question is rational. Exercise 1.2 breaks that assumption deliberately, asking what a number looks like when its decimal never settles into any repeating pattern at all. The decimal-classification instinct sharpened here also reappears in Class 10's decimal expansion exercise, where the same 2^m × 5ⁿ rule is proved rather than just applied.

One habit worth keeping from this exercise: whenever a fraction's decimal behaviour needs to be predicted rather than measured, factorising the denominator first is faster and less error-prone than dividing all the way out — division confirms the answer, but factorisation predicts it in a single glance.