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Level 9Grade 9skill: multi_step_word· 9 min read

Multi-Step Word Problems

In short: Multi-step word problems fail on reading, not arithmetic: every one hides a middle value the question never names. Find that value first, let the story set the order of operations (each step acts on the running total, not the original), and refuse the clean one-step shortcut — averaging speeds, adding percentages, or stopping early — because that is exactly the wrong answer built to catch you.

A repeatable method for multi-step word problems

Multi-step problems are not harder arithmetic — they are harder reading. The numbers are usually easy. What trips strong students is that the question asks for a quantity the problem never names out loud. There is always a hidden middle value you must compute before you can touch the final answer.

So work backwards from what is asked. If a question wants a difference, you need two things to subtract — find both first. If it wants a final price, ask what the price was one step earlier. Naming that hidden quantity is the whole game. Once you have it, the last step is almost trivial.

One warning that runs through every example below: when changes happen one after another, each change acts on the previous result, not on the original number. A second discount comes off the already-discounted price. Tax is charged on the reduced total. Get the order wrong and you compute a real number — just not the one that was asked for.

  1. Read the question line last: circle the exact quantity being asked for, and note whether it is a total, a difference, a final value, or a count.
  2. Name the hidden intermediate step. Write it as its own labelled number before doing anything else.
  3. Decide the order of operations from the story, not from left to right. Ask: what happens first, and does each step act on the original amount or the running total?
  4. Compute the intermediate value, label it with units, and only then do the final step.
  5. Sanity-check the size and sign of the answer against the story before you write it down.

The four patterns behind almost every multi-step trap Why it works

Successive changes
Each percentage or step acts on the result of the last step, not on the starting number. Chain by multiplying factors.

$80, 25% off then 8% tax = 80 × 0.75 × 1.08 = $64.80. Never 80 × (1 − 0.25 + 0.08).

Percent vs. fixed amount
When you mix a percentage discount with a fixed dollar-off, the order changes the answer. To pay least, take the percentage off the larger amount first.

$50 item, '20% off' and '$10 off'. Percent first: 40 − 10 = $30. Dollar first: 40 × 0.8 = $32. Order is worth $2.

Average of rates
You cannot average two speeds (or rates) directly. Average = total distance ÷ total time. Equal distances at different speeds always pull the average toward the slower one.

60 km at 40 km/h then 60 km at 60 km/h: 120 ÷ (1.5 + 1) = 120 ÷ 2.5 = 48 km/h, not 50.

Repeating cycles
Two events that repeat on periods a and b coincide every LCM(a, b). To count how often in a window, divide the window by the LCM — do not divide by a or b.

Flashes every 12 s and 18 s: together every LCM = 36 s. In 180 s that is 180 ÷ 36 = 5 times.

Notice the through-line: in every pattern the trap is a plausible one-step shortcut — add the percentages, average the speeds, divide by the smaller period. The shortcut always produces a clean number, which is exactly why it is dangerous. Treat these four as the core, not the whole list: the worked examples and mistakes below extend them with two close cousins — stopping at an intermediate value when a difference was asked for, and flipping the sign of a step. Slow down for the hidden step and the traps disappear.

Worked examples

Sam drives 60 km to a town at 40 km/h, then returns along the same road at 60 km/h. What is the average speed for the whole trip? = 48 km/h. (Averaging the speeds gives 50 km/h — tempting, and wrong, because Sam spends more time at the slower speed.)
  1. The hidden quantities are the two times, not a single average speed. Find each time first.
  2. Time out = 60 ÷ 40 = 1.5 hours.
  3. Time back = 60 ÷ 60 = 1 hour.
  4. Total distance = 120 km; total time = 1.5 + 1 = 2.5 hours.
  5. Average speed = 120 ÷ 2.5 = 48 km/h.
A backpack lists at $50. You hold two coupons, '20% off' and '$10 off,' and the store lets you choose the order. What is the lowest possible price, and how much does the ordering save you? = $30, and choosing the better order saves $2.
  1. This is a percent-versus-fixed mix, so order matters. Compute both orders.
  2. Percentage first: 50 × 0.80 = 40, then 40 − 10 = 30.
  3. Dollar-off first: 50 − 10 = 40, then 40 × 0.80 = 32.
  4. Lowest price is $30, taking the 20% off first.
  5. Saving from choosing well = 32 − 30 = $2.
You deposit $2,000 at 10% per year. How much more interest does compound interest earn than simple interest over 2 years? = $20 more. (Stopping at $420 answers a different question — that is the compound interest, not the difference.)
  1. This is a difference (a partial-computation trap): you need both interest amounts first — do not stop at one.
  2. Compound: 2000 × 1.10² = 2000 × 1.21 = 2420, so interest = 2420 − 2000 = $420.
  3. Simple: 2000 × 0.10 × 2 = $400.
  4. Difference = 420 − 400 = $20.
Two lighthouses flash together exactly at noon. One flashes every 12 seconds, the other every 18 seconds. In the next 3 minutes, how many more times do they flash together? = 5 times. (Counting the 3-minute mark itself gives 5; treating the window as open-ended and excluding t = 180 s would give 4 — so read the endpoint carefully.)
  1. They coincide every LCM(12, 18) seconds. Find the LCM first.
  2. 12 = 2² × 3 and 18 = 2 × 3², so LCM = 2² × 3² = 36 seconds.
  3. The window is 3 minutes = 180 seconds; count the 3-minute mark itself (t = 180 s) but not noon (t = 0).
  4. Count of coincidences = 180 ÷ 36 = 5 (at 36, 72, 108, 144, and 180 s).

Four ways these problems go wrong

Stopping at the intermediate value

✗ $420

On the interest problem, a student computes the compound interest, $420, and writes it as the answer. But the question asked how much more compound earns than simple — a difference, which needs two numbers, not one.

Fix: Re-read the final line before you commit. If it says 'difference,' 'how much more,' or 'left over,' you are not done until you have subtracted. Circle the target quantity in step 1 so you cannot forget it.

partial_computation

Applying steps in the wrong order

✗ $32

With the '20% off' and '$10 off' coupons, taking the $10 off first (50 − 10 = 40) and then 20% (40 × 0.8 = 32) gives $32. Reading the coupons left to right is not the same as the cheapest order.

Fix: Let the story, not the page order, set the sequence. When a percentage and a fixed amount mix, apply the percentage to the larger amount first to pay the least — and always check whether swapping the order changes the number.

order_of_ops_lr

Averaging rates instead of dividing totals

✗ 50 km/h

On the round trip, (40 + 60) ÷ 2 = 50 km/h feels right, but it uses the wrong operation. Speed is total distance over total time, and Sam spends 1.5 hours slow versus 1 hour fast, so the honest average is 48.

Fix: Never average two speeds directly. Rebuild the fraction: add the distances, add the times, then divide. The same rule covers price-per-unit, workers-per-job, and any 'per' quantity.

wrong_operation

Flipping a sign mid-calculation

✗ −$5 (a $5 credit)

Suppose you owe $75, pay back $50, then borrow $30 more. The correct debt is 75 − 50 + 30 = $55. Writing the new borrowing as another subtraction, 75 − 50 − 30 = −5, turns a $55 debt into a phantom $5 credit.

Fix: Decide the sign of each amount from the story before you chain them: paying back reduces debt (−), borrowing increases it (+). Say the direction out loud for each step and check the final sign makes sense.

sign_error

Practice

Try these multi-step problems. For each one, name the hidden intermediate value first, then decide the order of operations from the story before you compute. If a distractor tempts you, check which single-step shortcut it came from.

Related skills

First, make sure of

Frequently asked questions

When I have a percentage discount and a fixed dollar coupon, which should I apply first?

To pay the least, apply the percentage discount first, while the price is still large, then subtract the fixed amount. On a $50 item, '20% off' then '$10 off' gives $30, but reversing the order gives $32. The order genuinely changes the price whenever you mix a percentage with a fixed amount.

Why isn't the average speed just the average of the two speeds?

Because you spend different amounts of time at each speed. Over equal distances, you spend longer at the slower speed, so the true average is pulled toward it. Always rebuild it as total distance divided by total time: 60 km at 40 km/h and 60 km at 60 km/h gives 120 ÷ 2.5 = 48 km/h, not 50.

How do I spot the hidden step that these problems always seem to have?

Read the final question first and identify exactly what it asks for — a total, a difference, a final value, or a count. Then ask what number you would need to have in hand to answer it. That needed number is the hidden step. Compute and label it before touching the final calculation.

What's a quick way to compare compound and simple interest?

Simple interest is principal × rate × years. Compound interest is principal × (1 + rate)years, minus the principal. Over 2 years at 10% on $2,000, that is $400 versus $420 — a $20 gap. The gap comes entirely from earning interest on the first year's interest, and it grows with time.

For events that repeat on different cycles, how do I find when they line up?

Two events with periods a and b coincide every LCM(a, b). To count coincidences within a time window, divide the window by the LCM — never by a or b alone. Flashes every 12 s and 18 s line up every 36 s, so in 180 s (counting the 180 s mark itself) they coincide 180 ÷ 36 = 5 times.