Entrance exams: word problems

A word problem is a translation

The third pillar: word problems. For a lot of year 9s the most feared part of the test. And yet the secret is simple: a word problem is not new maths. It is a translation — from English into an equation. The maths inside, you already know.

The translation has its dictionary. “Five more” = +5. “Five times as much” = ·5. “Together” = a sum. “Left over” = subtracting. “They split in a ratio” = a ratio. “How many percent” = percents.

And it has a method that works on every problem: read twice → list what you know → label what you look for → set up an equation → solve → check against the question → answer in a sentence.

Underline that last step. Tests ask specifically: “How much did Ema pay?” If you calculate x correctly, but x was the price per item and they asked for the total spend, you do not get the mark. The last look always belongs to the question.

Today we will go through the method on three types that keep coming back in tests: shopping, age and shared work.

Filip reads a test question about age: “A father is three times older than his son, together they are 48 years…” and panics. Ema stops him: “Do not read it as a story, read it as a code. Son = x. Father = 3x. Together = plus. Equation: x + 3x = 48.” Filip finishes: “4x = 48, son 12, father 36.” He checks: 36 is three times 12, sum 48. “It almost translated itself.” Ema nods: “A word problem is a dictation in a foreign language. You already know the words.”

A translation method for word problems

Step 1: Read twice. The first time for the story, the second for the numbers. On the second reading underline the facts and the question.

Step 2: List the known and label the unknown. A short key: “price of a notebook = x, price of a pencil case = x + 60”. Always a key, always needed.

Step 3: Translate the sentences into an equation. Dictionary: “n more” = +n, “n times as much” = ·n, “together/in total” = a sum, “n less” = −n, “the same” = equals.

Step 4: Solve the equation. Classic: brackets, moving, divide. Line by line.

Step 5: Check against the QUESTION. Put the result into the text, not into your equation. “备注book 20, pencil case 80, together 100, pencil case 60 more expensive” — do both sentences match? Only now it is 完成.

Step 6: Answer the question that was asked, in a sentence, with a unit.

A test bonus — shared-work problems: “Painter A paints a fence in 6 hours, painter B in 3 hours. How long together?” The trick: in an hour A does a sixth of the fence, B a third. Together in an hour: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 of the fence. The whole fence therefore in 2 hours. Calculate the work per hour — and never average the times (4.5 hours is wrong: together it cannot take longer than the faster one alone!).

Example 1: shopping (an equation from prices)

Question: A T-shirt and a hoodie together cost 840 Kč. The hoodie is three times as expensive as the T-shirt. How much does the hoodie cost?

Method: Key: T-shirt = x (the smaller quantity), hoodie = 3x.

Equation: x + 3x = 840, so 4x = 840, so x = 210.

Hoodie: 3 · 210 = 630 Kč.

Check against the question: 210 + 630 = 840 — it matches. Hoodie three times as expensive — it matches.

Watch the question: they asked for the HOODIE, not the T-shirt. Answer: The hoodie costs 630 Kč. (Who answers 210 calculated correctly and still lost a mark.)

Example 2: age in a few years

Question: Tereza is 14, her mum 42. In how many years will mum be exactly twice as old as Tereza?

Method: Key: the sought number of years = x.

In x years: Tereza 14 + x, mum 42 + x.

Equation (“twice as old”): 42 + x = 2 · (14 + x).

I expand: 42 + x = 28 + 2x. I move: 42 − 28 = 2x − x, so x = 14.

Check: In 14 years Tereza will be 28 and mum 56. And 56 = 2 · 28. It matches.

Answer: In 14 years. With age problems do not forget to add the years to BOTH — the question counts on that.

Example 3: shared work

Question: One pump empties a pool in 4 hours, the second in 12 hours. How long will they empty it together?

Method: Work per hour: the first 1/4 of the pool, the second 1/12 of the pool.

Together per hour: 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3 of the pool.

The whole pool: 3 hours.

Sense check: 3 hours is less than 4 (the faster pump alone) — a helper must shorten the time. It matches.

Answer: Together they empty the pool in 3 hours. The average of the times (8 hours) would be nonsense — a sense check kills it at once.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

A mix like in entrance exams — each question different, like in a real test. Write fractions as a/b, decimals with a comma.

Practise the questions

On paper

Four word problems, for each the whole translation method including an answer in a sentence.

1. A book and a bookmark together cost 275 Kč. The book is 245 Kč more expensive than the bookmark. How much does the bookmark cost? (A classic — do not rush, the result surprises.)

2. Adam is 15, his sister 9. In how many years will Adam be exactly one and a half times as old? Set up an equation and do the check.

3. One printer prints flyers in 6 hours, the second in 3 hours. How long together? Check with sense.

4. Tickets for 2 adults and 3 children cost 620 Kč. A child’s is 60 Kč cheaper than an adult’s. How much does an adult ticket cost? (Two unknowns — remember simultaneous equations from lesson 7.)

Now you 💪

1. You underline facts and the question in the problem on the second reading.
2. You give the unknown to the smallest quantity and build the equation from the dictionary.
3. You check against the text of the question and answer in a sentence to the question that was asked.

What to take from this lesson

• A word problem is a translation from English into an equation. Words: “more” = +, “times as much” = ·, “together” = a sum.
• The unknown belongs to the smallest quantity — the equation comes out without fractions.
• With age add years to everyone, with shared work add the work per hour.
• The check is 完成 against the question, not against your own equation.
• The last look belongs to the question: answer what they asked.

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