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Problem Solving
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Identifying operations
Identify whether to add or subtract
\[ \text{I have £10 and spend £3.50…} \]
Recognise which operation is needed from problem context.
Identify whether to multiply or divide
\[ \text{8 pens cost £12. One pen = ?} \]
Recognise when multiplication or division is needed.
Identify the operations in a two-step problem
\[ \text{Buy 3 drinks, get change from £10} \]
Recognise the sequence of operations in a multi-step problem.
Two-step: add then subtract
Find total cost then calculate change
\[ \text{Book + Pen, change from £10?} \]
Add items first then subtract from payment.
Find total cost of three items then change
\[ \text{Sandwich + Drink + Crisps, change?} \]
Add three items then calculate change.
Two-step: multiply then subtract
Find cost of multiple items then change
\[ \text{6 apples at 40p, change from £5?} \]
Multiply to find total then calculate change.
Find cost of multiple items then remaining budget
\[ \text{£25, buy 4 books at £3.50, left?} \]
Multiply for total spending then subtract from initial amount.
Two-step: add then multiply
Find combined cost then multiply
\[ \text{Coffee £2.50 + Cake £3, cost of 2 each?} \]
Add item prices then multiply by quantity.
Two-step: multiply then add
Find cost of different quantities then total
\[ \text{3 coffees at £2.80 + 2 cakes at £3.50} \]
Multiply each item by its quantity then add.
Two-step: subtract then multiply
Find unit savings then total savings
\[ \text{Was £25, now £18. Save on 3 shirts?} \]
Find saving per item then multiply.
Two-step: add then divide
Find total then split equally
\[ \text{Amy £8.50 + Ben £11.50, split equally?} \]
Add amounts then divide by number of people.
Two-step: multiply then divide
Find total cost then cost per person
\[ \text{4 pizzas at £9, split by 3 people?} \]
Calculate total then divide among people.
Budget problems
Check if budget is sufficient
\[ \text{£15 budget – enough for items?} \]
Calculate total and compare to budget.
Find how much more is needed
\[ \text{Have £8, want £12.50 game. Need?} \]
Calculate shortfall from a budget.
Find how much would be left
\[ \text{£20 – £7.30 – £5.45 = ?} \]
Calculate total spending and subtract from budget.
Find maximum affordable with constraint
\[ \text{£30 budget, £8.50 tickets. Max?} \]
Find maximum quantity within budget.
Bills and receipts
Calculate total from itemised bill
\[ \text{Burger + Chips + Drink = ?} \]
Add items on a simple bill.
Calculate total with quantities shown
\[ \text{2 × Coffee + 1 × Cake + 3 × Biscuit} \]
Calculate total when quantities are given.
Verify a bill total
\[ \text{Items sum to £10.00 – correct?} \]
Check if a bill total is accurate.
Find the error in a bill
\[ \text{Stated total wrong. Correct total?} \]
Correct an incorrect bill total.
Find missing item price from total
\[ \text{Total £18.40, Starter + Main = ? Dessert?} \]
Subtract known items from total.
Add a fixed service charge
\[ \text{Food £52 + Service £6.50 = ?} \]
Add a service charge to a subtotal.
Unequal splits
Split a bill in ratio 2:1
\[ \text{£27 split, Jo pays twice Kim} \]
Divide according to a simple ratio.
Find each share when split unequally
\[ \text{£40 prize, Ali gets £10 more than Ben} \]
Find shares given a difference constraint.
Working backwards
Find starting amount from spending and remainder
\[ \text{Spent £14.50, £8.50 left. Started?} \]
Add spending and remainder to find initial amount.
Find cost from quantity and budget left
\[ \text{5 items, £3.50 left from £20. Each?} \]
Work backwards to find unit cost.
Find quantity from spending and unit cost
\[ \text{Spent £21 on £3.50 books. How many?} \]
Divide total spend by unit cost.
Comparison problems
Find how much more one item costs than another
\[ \text{Laptop £425, Tablet £289. Difference?} \]
Calculate the difference between two prices.
Compare total spending of two people
\[ \text{Anna: 3 × £4, Ben: 2 × £5.50. Who more?} \]
Calculate and compare two people’s spending.
Compare two deals
\[ \text{Shop A: 4 for £5, Shop B: 5 for £6} \]
Compare total cost at different shops.
Saving problems
Calculate total saved over time
\[ \text{£3.50/week for 8 weeks = ?} \]
Multiply weekly savings by number of weeks.
Find how long to save for a target
\[ \text{Bike £84, save £7/week. How long?} \]
Divide target by savings rate.
Find weekly savings needed for a target
\[ \text{Save £60 in 12 weeks. Per week?} \]
Divide target by time period.
Special cases
Problem with zero answer
\[ \text{Have £5.50, spend £5.50. Left?} \]
Recognise when the answer is zero.
Problem with answer under £1
\[ \text{3 friends share 75p equally} \]
Handle answers in pence.
Problem mixing pence and pound notation
\[ \text{80p + £1.50 = ?} \]
Add amounts given in different formats.
Timer (Optional)
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Question