TARA Problem Solving practice questions

A cinema charges the ticket prices shown below.

On a Saturday, Mr and Mrs Okafor go to the cinema with their son, aged 15, who has a student card; their daughter, aged 12; and Mrs Okafor's father, aged 70.

What is the least they can pay for their five tickets?

The six-digit code for Tom's front door is 4 7 7 2 4 7.

His neighbour says her code follows exactly the same pattern of repeated digits: wherever two digits in Tom's code are equal, the digits in the same positions of her code are equal, and wherever two digits in Tom's code are different, the digits in those positions of her code are different too.

Which of the following could be her code?

A small bookshop sells every fiction book for £8, every children's book for £5 and every non-fiction book for £10. The table shows how many books of each kind it sold on each day last week.

On which day did the shop take the most money from book sales?

Starting outside the town hall, Mia walks 400 m north. She turns left and walks 300 m, turns left again and walks 800 m, then turns left once more and walks 600 m.

Where is she now, compared with the town hall?

Nina went shopping. She spent one third of her money on a jacket, then half of what she had left on a pair of shoes, and then £12 on lunch. She came home with £18.

How much money did she have when she set out?

Ellie's party punch is made from 2 litres of orange juice, 3 litres of apple juice and 1 litre of lemonade.

Which of the following batches has exactly the same proportions, and so would taste the same?

Priya needs to be in Corley by 09:30.

Trains from Ashby to Brenton leave at 07:12 and then every 25 minutes, and the journey takes 38 minutes. Buses from Brenton station to Corley leave at 5 minutes and 35 minutes past each hour, and the journey takes 22 minutes. Priya needs at least 5 minutes to get from the train to the bus.

What is the latest train from Ashby that she can catch?

Hana is posting two parcels, using the price list below. Prices depend on the size of the box and the total weight of the parcel, including packaging.

The first parcel holds a book weighing 0.9 kg in a box measuring 30 cm × 22 cm × 8 cm. The second holds a lamp weighing 1.9 kg in a box measuring 50 cm × 30 cm × 30 cm. Each box, with its packing material, weighs 0.2 kg.

How much will Hana pay to post both parcels?

A café's lunch deal is one main, one side and one drink.

Mains: soup, pasta, curry or a wrap. Sides: bread, chips or salad. Drinks: tea or coffee (hot); cola, juice or water (cold).

Soup is always served with bread, so anyone choosing soup must have bread as their side. Hot drinks are not offered with salad.

How many different lunch deals are possible?

The table shows the number of members of a sports club in each age group in 2024 and 2025.

Which age group had the largest percentage increase in members from 2024 to 2025?

Sam, who is 23, wants to hire a car for 4 days and expects to drive 500 miles in total. He has three quotes, shown in the table.

What is the cheapest price Sam can get for the hire?

Five runners, Ana, Ben, Cal, Dee and Eve, finished a race, with no two finishing together.

Ben finished ahead of Eve but behind Dee. Cal finished immediately behind Ana. Ana did not win. Eve did not finish last.

Who finished third?

1 March 2027 is a Monday.

Which of the following months of 2027 also begins on a Monday?

A solid wooden block measuring 5 cm × 4 cm × 3 cm stands on a table on one of its 5 cm × 4 cm faces. The top and the four sides are painted; the face resting on the table is not. The block is then cut into 1 cm cubes.

How many of the small cubes have no paint on them at all?

A town car park has these rules. The first 30 minutes of any stay are free. Time after the first 30 minutes is charged at £1.20 for each hour or part of an hour. Nobody pays more than £6.00 for a stay that starts and ends on the same day. Blue Badge holders park free.

One Saturday, Ken parks from 09:40 to 14:05 and his sister parks from 08:15 to 17:50. Their mother, who has a Blue Badge, parks from 10:00 to 12:00. Neither Ken nor his sister has a Blue Badge.

How much do the three of them pay altogether?

Six teams take part in a tournament in which every team plays every other team exactly once. A win earns 3 points, a draw earns 1 point for each of the two teams, and a loss earns nothing.

At the end of the tournament, the six teams' points add up to 40.

How many matches were drawn?

A shop sells the same pen in three ways: single pens at 80p each, packs of 3 for £2.00, and packs of 5 for £3.00.

Leo needs at least 22 pens for his class. What is the least he can spend?

The Patel household is comparing two electricity tariffs.

Tariff A: a standing charge of 40p per day, plus 24p for each kWh used. Tariff B: no standing charge, and 30p for each kWh used.

The table shows their electricity use over three months.

Over these three months together, which tariff would have cost less, and by how much?

Two decorators are refurbishing a flat. The table lists the jobs, how long each takes, and which jobs must be finished before each one can start.

Each job is done by one decorator from start to finish (it cannot be shared or split), and each decorator can do only one job at a time.

What is the shortest time in which all the jobs can be completed?

A goods lift can carry at most 1000 kg on each journey. Six crates must be taken up to the top floor. They weigh 600 kg, 550 kg, 550 kg, 500 kg, 450 kg and 300 kg. The crates cannot be opened or split.

What is the least number of journeys the lift must make to take all the crates up?

On an ordinary dice, opposite faces add up to 7. A dice is placed on a table with 1 on top, 2 facing north and 3 facing east.

The dice is tipped over one of its bottom edges four times: first to the east, then to the north, then to the east again, and finally to the south. Tipping it to the east means rolling it over its eastern bottom edge, so that the face that was on top ends up facing east; the other directions work the same way.

Which number is now on top?

A digital clock shows the time in 24-hour format, from 00:00 to 23:59.

How many times each day does the four-digit display read the same forwards and backwards (for example, 13:31)?