You may know how to solve an equation but struggle to create one from a paragraph. The difficulty often comes from deciding what the unknown represents and how each sentence describes it.
Do not begin by searching for a formula at random. Identify the requested quantity, name the unknown with its units and translate one relationship at a time. Then check whether the resulting equation describes the original situation.
This guide uses original practice examples for O Level Mathematics. Work alongside your current syllabus and use the methods requested in each question. For the wider subject plan, see Abeel’s O Level Mathematics guide.
Define the unknown precisely
“Let x be the answer” is usually less useful than “let x be the number of adult tickets sold.” The second definition tells you what x measures and whether fractional or negative values would make sense.
If x is a length, state the unit. If it is a price, say whether the amount is in rupees or another currency. If it represents time, distinguish hours from minutes.
A precise definition makes later expressions easier to interpret. If x counts adult tickets out of fifty tickets in total, the number of remaining student tickets is 50 − x.
Avoid changing the meaning of x halfway through. Once it represents a count, do not reuse it for a ticket price in the same solution without clearly introducing a new variable.
Separate facts from the requested result
Read the question once for the situation and again for the mathematical relationships. Mark the totals, differences, rates, percentages and constraints that actually connect the quantities.
Then identify what the final answer must state. A problem may ask for the rectangle’s length even though you choose its width as the unknown. Solving x is only an intermediate step in that case.
Not every number in a paragraph belongs in the same equation. Some information may describe a later part or provide context. Explain the role of each number before using it.
Write a short verbal relationship first if needed: “total revenue equals adult revenue plus student revenue.” The equation should express that sentence, not merely contain the numbers you noticed.
Translate comparison phrases in the correct order
“Five more than twice x” means 2x + 5. “Five less than twice x” means 2x − 5. By contrast, “x less than five” means 5 − x.
Subtraction order matters. A phrase involving “less than” does not always put the first quantity first in the final expression.
Test the phrase with a simple numerical value. If x = 3, five more than twice x is eleven. The expression 2(x + 5) would give sixteen, describing a different operation.
Brackets communicate which operation happens first. “Twice the sum of x and five” is 2(x + 5), while “the sum of twice x and five” is 2x + 5.
Use a rectangle to distinguish perimeter and area
Suppose a rectangle’s length is four centimetres more than its width, and its perimeter is forty centimetres. Let the width be x centimetres, making the length x + 4 centimetres.
Perimeter adds all four sides, so 2x + 2(x + 4) = 40. Simplifying gives 4x + 8 = 40, so x = 8.
The width is eight centimetres and the length is twelve centimetres. Their perimeter is 2(8) + 2(12) = 40, confirming the given total.
The product x(x + 4) would describe area, not perimeter. In this example, area is 8 × 12 = 96 square centimetres, but that is not the equation supplied by the perimeter statement.
Represent consecutive quantities systematically
If three consecutive odd integers are required, you can write them as x, x + 2 and x + 4, with x an odd integer. Their separation is two, not one.
Suppose their sum is thirty-nine. The equation is x + (x + 2) + (x + 4) = 39. This gives 3x + 6 = 39 and x = 11.
The integers are eleven, thirteen and fifteen. Their sum is thirty-nine and each is odd, so both the arithmetic and the description are satisfied.
Writing x, 2x and 3x would describe multiples of a quantity rather than consecutive odd integers. Check the relationship between the proposed expressions before solving the resulting equation.
Express a percentage as a multiplier
An item costs PKR 340 after a fifteen per cent discount. Let the original price be p rupees. After the discount, eighty-five per cent of the original remains, so 0.85p = 340.
Dividing gives p = 400. Check by calculating fifteen per cent of four hundred, which is sixty, and subtracting it to obtain three hundred and forty.
Adding fifteen per cent of the discounted price does not generally recover the original price. That would use a different base for the percentage calculation.
Label the base quantity in every percentage problem. A percentage increase, percentage decrease and percentage of a new amount can look similar while describing different equations.
Use totals to express a second unknown
Imagine fifty tickets are sold. Adult tickets cost PKR 200, student tickets cost PKR 120, and total revenue is PKR 8,400. These are invented figures for practice.
Let a be the number of adult tickets. There are 50 − a student tickets. The revenue equation is 200a + 120(50 − a) = 8400.
Expanding gives 200a + 6000 − 120a = 8400, so 80a = 2400 and a = 30. Therefore twenty student tickets were sold.
Check both relationships: thirty plus twenty equals fifty, and 30 × 200 + 20 × 120 equals 8,400. Checking only the revenue would miss a possible error in the total ticket count.
Translate a ratio using a shared multiplier
Suppose two quantities are in the ratio three to five and their total is sixty-four. Represent them as 3k and 5k rather than as three and five themselves.
The total gives 3k + 5k = 64, so k = 8. The quantities are twenty-four and forty.
The ratio has eight equal parts in total. Each part is worth eight in this example, which explains the shared multiplier.
If the question instead gives the difference between the quantities, the equation changes. A difference of sixteen would give 5k − 3k = 16. Identify whether the stated number is a total, a difference or one of the individual quantities.
Make units consistent in rate problems
If a journey covers 120 kilometres in two and a half hours, average speed is distance divided by time: 120/2.5 = 48 kilometres per hour.
Thirty minutes is half an hour, not 0.30 hours. A correct-looking rate formula can give a wrong answer when time is converted incorrectly.
Name the rate’s units before substituting. If distance is in metres and time in seconds, the resulting speed is metres per second. Convert either before or after calculation in a clearly stated way.
Do not assume a constant speed merely because you can calculate an average speed. A contextual problem may distinguish sections of a journey, waiting time or changing rates, requiring separate relationships.
Use an inequality when the wording gives a limit
A service has a fixed charge of PKR 500 plus PKR 20 for each unit used. A customer wants the total to be no more than PKR 900.
Let m be the number of units. The condition is 500 + 20m ≤ 900, giving m ≤ 20. If only whole non-negative units are possible, the allowed counts run from zero through twenty.
Using an equals sign alone would identify the boundary cost but would not express all amounts within the budget. Words such as “at most,” “at least,” “more than” and “less than” identify different inequalities.
If solving involves multiplying or dividing by a negative number, reverse the inequality direction. Show that step carefully rather than applying equation rules without checking the relation.
Solve only after the equation has been checked
Before carrying out several algebraic steps, read each term back into the context. In the ticket equation, 200a is adult revenue and 120(50 − a) is student revenue. Both are money amounts, so their sum can equal total revenue.
An equation that adds a number of tickets directly to a price is likely mixing incompatible quantities. Units provide a useful model check.
Draw a small diagram or table when the relationship is difficult to hold in your head. A rectangle diagram can label sides; a ticket table can separate count, price and revenue.
Then solve using valid algebra and keep enough working to trace the method. Solving an incorrect equation accurately does not answer the original question.
Interpret roots and reject values for a stated reason
Some word problems lead to a quadratic with more than one algebraic solution. Check each solution against the original context and any restrictions.
A negative length is not a valid physical dimension in an ordinary geometry problem. A fractional count of indivisible tickets may also be impossible. State the reason rather than discarding an unfamiliar answer without explanation.
Do not reject every negative value automatically. Temperature, relative position or financial change can legitimately be negative depending on the definition of the variable.
If no solution fits the context, check the equation, arithmetic and transcription. Do not round an impossible result into a plausible-looking answer unless approximation is actually justified by the question.
Review modelling errors separately from algebra errors
When marking an attempt, ask whether the equation was correct before examining the solution steps. A correct model with a sign error needs different practice from an incorrect model solved perfectly.
Keep a short record of the phrase that caused difficulty, its correct mathematical translation and one fresh example. Practise building equations without solving them for a few minutes, then return to complete problems.
Use Abeel’s Cambridge past-paper guide and O Level revision plan to include these tasks in a wider routine.
Explore Abeel Eschool and current courses for subject options. Ask through admissions and FAQs, sharing your variable definition and proposed equation so the difficult step is clear.
Frequently asked questions
What should I write before forming an equation?
Define the unknown precisely, including its unit or meaning. Then state the relationship that connects it to the given information.
How do I know whether to use one or two variables?
If a total lets you express the second quantity in terms of the first, one variable may be convenient. Otherwise two variables and two independent relationships may be clearer.
Why can I not reverse a discount by adding the same percentage?
The discounted price is a different base. Use the multiplier relating the original price to the final amount.
When should I use an inequality?
Use one when the wording gives a limit or range, such as no more than, at least or strictly less than.
How do I check a word-problem answer?
Substitute into every original relationship, check units and verify that the values satisfy the context’s restrictions. Do not check only the final algebraic line.
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