Multi-Step Problem Solving

Many UCAT Quantitative Reasoning questions require you to combine multiple calculation steps to arrive at the answer. These are often the most challenging questions because you must identify which steps are needed, extract the correct data, perform the calculations in the right order, and avoid errors at each stage.

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What Makes a Question Multi-Step?

A multi-step question requires two or more separate calculations to reach the final answer. Each step may use a different operation or different data from the data set.

Example of a Single-Step Question

"What is 15% of 2,400?" → One calculation: 0.15 × 2,400 = 360.

Example of a Multi-Step Question

"A shop sells 150 items at £12 each and 80 items at £18 each. What is the total revenue?" → Two calculations (150 × 12 and 80 × 18) plus an addition.

"If this total revenue represents a 20% increase on last year, what was last year's revenue?" → Now there is a third step (reverse percentage).

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The Step-by-Step Strategy

Step 1: Read the Question and Identify the End Goal

What is the question ultimately asking for? Start with the end in mind.

Step 2: Work Backwards to Identify Intermediate Steps

What do you need to calculate to get to the end goal? And what do you need before that?

Step 3: Identify the Data You Need

For each step, what values do you need from the data set? Locate them precisely.

Step 4: Calculate Each Step

Work through each calculation carefully. Write intermediate results on the whiteboard.

Step 5: Check the Units and Reasonableness

Does your final answer make sense? Are the units correct?

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Worked Example 1: Revenue and Profit

Data:

Product	Cost Price	Selling Price	Units Sold
Alpha	£4.50	£7.80	320
Beta	£3.20	£5.50	480
Gamma	£6.00	£9.40	150

Question: What is the total profit from selling all three products?

Solution

Step 1: Profit per unit = Selling Price - Cost Price

Product	Profit per Unit
Alpha	£7.80 - £4.50 = £3.30
Beta	£5.50 - £3.20 = £2.30
Gamma	£9.40 - £6.00 = £3.40

Step 2: Total profit per product = Profit per Unit × Units Sold

Product	Total Profit
Alpha	£3.30 × 320 = £1,056
Beta	£2.30 × 480 = £1,104
Gamma	£3.40 × 150 = £510

Step 3: Total profit = £1,056 + £1,104 + £510 = £2,670

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Worked Example 2: Compound Problem with Percentages

Data:

A clinic sees 1,200 patients per month. Of these:

• 40% are follow-up appointments
• 35% are new patient appointments
• The remainder are emergency walk-ins
• Follow-up appointments last 15 minutes on average
• New patient appointments last 30 minutes on average
• Emergency walk-ins last 20 minutes on average

Question: What is the total number of appointment-minutes for emergency walk-ins per month?

Solution

Step 1: Calculate the percentage of emergency walk-ins. 100% - 40% - 35% = 25%

Step 2: Calculate the number of emergency walk-ins. 25% of 1,200 = 0.25 × 1,200 = 300 patients

Step 3: Calculate total minutes. 300 × 20 = 6,000 minutes

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Worked Example 3: Rate-Based Problem

Data:

Task	Worker A (items/hour)	Worker B (items/hour)
Packaging	45	35
Labelling	60	55
Inspection	30	25

Question: Worker A packages items for 3 hours and Worker B labels items for 4 hours. How many items do they process in total?

Solution

• Worker A packaging: 45 × 3 = 135 items
• Worker B labelling: 55 × 4 = 220 items
• Total: 135 + 220 = 355 items

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The "Which of the Following" Elimination Approach

Some multi-step questions present four answer options and ask which is correct. Rather than calculating the exact answer, you can sometimes eliminate wrong options using estimation.

Worked Example