Chapter 2.5 — Exercise 2.4 — More Applications

A Reliable Four-Step Method

• Assign a variable precisely. Rather than a vague "let x be the unknown," state exactly what it represents — "let x be the units digit" or "let x be Reshma's present age."
• Translate the condition into an equation. Take the key relationship described in words and rewrite it using the variable.
• Solve using transposition. This is the mechanical step, already practised in the earlier exercises of this chapter.
• Interpret the result. If x was only one piece of what's being asked — a digit, a ratio part, an age — compute the remaining quantities and state the full answer, not just the value of x.

Working Through the Eight Problems

Parallel lines (Q1): when lines l and m are parallel and cut by a transversal, alternate exterior angles are equal, so 2x + 15° = 3x − 10°, giving x = 25. A geometric fact becomes a one-line equation once the angle relationship is identified.

Straightforward number problems (Q2, Q5): "eight times a number reduced by 10 equals six times the number plus 4" becomes 8x − 10 = 6x + 4, giving x = 7. "Tripling a number and adding 2 gives the same result as subtracting the number from 50" becomes 3x + 2 = 50 − x, giving x = 12.

A two-digit number problem (Q3): if the digits sum to 9 and the units digit is x, the tens digit is (9 − x) and the number itself is 10(9 − x) + x. Subtracting 27 reverses the two digits, and solving that condition gives x = 3 and the original number 63.

A ratio-and-parts problem (Q4): a number splits into two parts in the ratio 5:3, with one part 10 more than the other. Letting the parts be 5x and 3x, the condition 5x = 3x + 10 gives x = 5, so the parts are 25 and 15, and the whole number is 40.

Age problems across time (Q6, Q7): in Q6, Mary is currently twice her sister's age, but in 5 years will be only 2 years older — 2x + 5 = x + 7 gives the sister's age as 2 years and Mary's as 4 years. In Q7, laying out present age, age-in-5-years, and age-9-years-ago side by side makes the relationship clear: x + 5 = 3(x − 9) gives x = 16.

A population changing by a percentage (Q8): the most demanding of the eight. Starting population x increases by 1200, then that new total decreases by 11%, landing 32 below the original x.

Why the Variable Choice in Q1 Matters

Q1's parallel-lines problem is worth a second look because it shows how a geometric fact does all the real work before any algebra starts. Alternate exterior angles being equal isn't something algebra can derive — it's a property of parallel lines cut by a transversal, established separately in geometry. Once that fact identifies which two angle expressions must be equal, the rest — 2x + 15° = 3x − 10°, giving x = 25 — is ordinary transposition. The pattern repeats across many geometry-flavoured word problems in this course: a non-algebraic fact supplies the equation, and algebra only takes over once that equation exists.

Setting Up the Problem Correctly, Not Just Solving It

• Age problems need a clear timeline. Keep "present age," "age after n years," and "age n years ago" visually distinct — writing them as a small table, as Q6 does, prevents mixing them up.
• Solve for everything the question asks, not just x. If a question asks for "the number and the two parts," compute 5x and 3x separately once x is known — the value of x alone isn't the full answer.
• A two-digit number is 10 × (tens digit) + (units digit), never simply the sum of its digits — this distinction is exactly what Q3 depends on.
• A percentage decrease means multiplying by (100 − rate)/100, not subtracting the rate directly from the total — an 11% decrease multiplies by 89/100, as Q8 shows.

Checking a Word-Problem Answer Against the Story, Not Just the Equation

Verifying a word problem's answer means more than confirming LHS = RHS in the equation — it means checking the result still makes sense against the original description. For Q3's two-digit number: the digits found are 3 (tens) and 6 (units)... but rereading the setup, the tens digit was defined as (9 − x) with x = 3, giving tens digit 6 and units digit 3, so the number is 63. Checking against the story directly: does 63 have digits summing to 9? Yes (6 + 3 = 9). Does subtracting 27 reverse its digits? 63 − 27 = 36, and 36 is indeed 63 with its digits swapped. Both conditions hold, confirming the answer independently of re-deriving the algebra — exactly the kind of check that catches a mislabelled variable even when the arithmetic itself was correct. The same two-step check — confirm the equation balances, then confirm the story still makes sense — is worth applying to every word problem in this exercise, not only the ones that feel complicated.

Why These Eight Problems Are Grouped Together

At first glance, a parallel-lines angle problem, a two-digit number puzzle, and a population changing by a percentage don't look like they belong in the same exercise. What ties them together is that in every case the described relationship is genuinely linear once it's written down — no squaring, no multiplying two unknowns together — even though the surface description (angles, digits, percentages) varies widely. Recognising that a situation reduces to a linear equation, despite not looking like one in plain language, is the actual skill this exercise is building, more so than any one problem type — and it's the skill that carries over to situations this exercise never explicitly covers, long after the specific angle or coin-counting problems here are forgotten — spotting the linear structure hiding inside a wordy description is a transferable habit, not a memorised list of problem types — and it's what makes an unfamiliar word problem feel approachable rather than intimidating, regardless of the specific scenario it's dressed up in.

Next Stop: Fractions

The equation-formation skill from this exercise continues into Exercise 2.5, where the same word-problem approach is applied to equations involving fractions. Age and number problems of this style reappear in linear equations in two variables in Class 9, while the ratio and percentage reasoning connects to Ratio and Proportion. For the bracket-expansion technique this exercise leans on, revisit Exercise 2.3.