Chapter 1.4 — Exercise 1.2 — Number Line

Question 1 — Plotting Single Points

Part (i): Represent 9/7 on the number line.
9/7 = 1 + 2/7, so its value sits strictly between 1 and 2. Since the denominator is 7, divide the segment from 1 to 2 into 7 equal parts. Counting 2 divisions to the right of 1 lands exactly on 9/7.

Part (ii): Represent −7/5 on the number line.
−7/5 = −1 − 2/5, so its value sits strictly between −1 and −2. With denominator 5, divide the segment from −1 to −2 into 5 equal parts, then count 2 divisions to the left of −1 to reach −7/5.

Question 2 — Multiple Points, One Number Line

Plot −2/13, 5/13 and −9/13 together. All three share denominator 13, which is what makes plotting them side by side straightforward: −2/13 and −9/13 both lie between −1 and 0, while 5/13 lies between 0 and 1. Divide both the (−1, 0) segment and the (0, 1) segment into 13 equal parts. Then −9/13 sits 9 divisions left of 0, −2/13 sits 2 divisions left of 0, and 5/13 sits 5 divisions right of 0 — giving the left-to-right order −9/13, −2/13, 0, 5/13.

Question 3 — Five Numbers Smaller Than 5/6

A quick way to generate rational numbers smaller than a given fraction with the same numerator is to increase the denominator — since 1/n shrinks as n grows, 5/n also shrinks. Keeping the numerator fixed at 5 and increasing the denominator past 6 gives: 5/7, 5/8, 5/9, 5/10, 5/11 — all strictly smaller than 5/6, and there are infinitely many more where these came from.

Question 4 — Twelve Rational Numbers Between −1 and 2

Convert both endpoints to a common denominator wide enough to fit at least twelve values in between. Using denominator 10: −1 = −10/10 and 2 = 20/10. Any twelve of the thirty integers strictly between −10 and 20 work, for example: −9/10, −7/10, −5/10, −3/10, −1/10, 0, 2/10, 5/10, 9/10, 11/10, 15/10, 19/10.

Question 5 — One Rational Number Between 2/3 and 3/4

Convert to a common denominator of 12: 2/3 = 8/12 and 3/4 = 9/12. Since these are consecutive twelfths with nothing between them yet, scale up by 2 to get 16/24 and 18/24 — now 17/24 sits cleanly between them.

Question 6 — Ten Rational Numbers Between −3/4 and 5/6

Convert to a common denominator of 12: −3/4 = −9/12 and 5/6 = 10/12. That leaves eighteen integers strictly between −9 and 10 to choose ten from, for example: −8/12, −7/12, −5/12, −3/12, −2/12, 0, 3/12, 5/12, 6/12, 8/12.

A Faster Route for "One Number Between Two Fractions"

Question 5's common-denominator method always works, but there's a quicker route: the average (mean) method. The number exactly halfway between any two rational numbers a and b is (a + b) ÷ 2, and it is always guaranteed to lie strictly between them.

Applying it to 2/3 and 3/4: (2/3 + 3/4) ÷ 2 = (8/12 + 9/12) ÷ 2 = (17/12) ÷ 2 = 17/24 — the same answer reached earlier by scaling the common denominator, but in fewer steps. The mean method is especially useful when a question only asks for one number in between and doesn't require the full list.

A Second Worked Example — Ordering Mixed Signs

A related question asks you to arrange rational numbers with different denominators in ascending order — for instance, −7/5, 9/7, −2/13 and 5/13. The number-line reasoning from Questions 1 and 2 answers this directly without needing a fresh method: −7/5 lies between −2 and −1, −2/13 lies between −1 and 0, 5/13 lies between 0 and 1, and 9/7 lies between 1 and 2. Reading positions left to right immediately gives the ascending order −7/5 < −2/13 < 5/13 < 9/7 — no separate "ordering" technique is needed once you can place each number on the line.

The One Idea Behind Every Part

• Between two integers: identify which pair of consecutive integers a fraction falls between by comparing it to the nearest whole numbers.
• Common denominator first: whenever the question involves more than one rational number — plotting several together, or finding numbers between two fractions — convert everything to like fractions before doing anything else.
• "Between two fractions, always more exist": there is no smallest gap between two distinct rational numbers; scaling to a larger common denominator always reveals more numbers in between. This is the density of rational numbers, and it's worth stating explicitly if a question asks how many rational numbers lie between two given values — the answer is always infinitely many.

Five Places a Sign or a Shortcut Goes Wrong

• Miscounting direction on negative numbers. For −7/5, the count of 2 divisions happens to the left of −1 (further from zero), not to the right — a frequent source of sign errors when plotting negatives.
• Assuming there's one "correct" answer. For questions asking for numbers between two values, any valid fraction in range is acceptable — your answer can differ from the textbook's and still be fully correct.
• Forgetting to check the sign of the whole expression after cross-multiplying two negative fractions — a double-negative slip is easy to make when converting −3/4 and 5/6 to twelfths.
• Stopping at the first common denominator found. If it turns out too narrow to fit as many numbers as the question asks for, scale both fractions up further (multiply numerator and denominator by the same larger factor) rather than assuming the question is unsolvable.
• Skipping the simplification check. An answer like 6/12 is perfectly valid but is more clearly checked against the original range once reduced to 1/2 — simplify before you double-check that a chosen value truly falls between the two given numbers.

Why the Diagram Matters, Not Just the Answer

Plotting a rational number is really a two-part statement: which unit segment it falls in, and how many equal parts that segment has been divided into. A point marked in roughly the right place without showing the division into q equal parts doesn't actually demonstrate that 9/7 has been located rather than guessed at. Get into the habit of stating both parts explicitly — "between 1 and 2, divided into 7 equal parts, at the 2nd mark" — since that sentence is the plotting method in words, and it's what makes the diagram a genuine construction rather than an estimate.

Fractions Today, Decimals Next

Comfortably converting to common denominators and reasoning about position on a line carries directly into Exercise 1.3, where the same numbers are handled in decimal form instead of fraction form. The density idea — that infinitely many rational numbers sit between any two others — reappears in Class 9 when Real Numbers introduces irrational numbers that fill in the remaining gaps on the line.