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Problem Solving

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Topic overview

Problem Solving is the second TARA module: 22 questions, 40 minutes, testing numerical reasoning, **requiring only basic mathematics, with no calculator permitted**. The official specification divides it into three skill types: relevant selection (picking out the useful numbers from the question stem), finding the procedure (finding the correct algorithm), and identifying similarity (applying an already-solved structure to new numbers). It emphasises logic and estimation, not laborious calculation.

Key concepts

The question stem often stuffs in irrelevant data as distraction. First ask 'which quantities do I actually need to answer this question?' Cross out the irrelevant ones, unify the units, then calculate. No calculator, so make good use of estimation to eliminate obviously wrong options first.

Key points

  • ◆First identify the quantity the question asks for, then find the data
  • ◆Unify the units (yuan/gram, km/h) before calculating
  • ◆Estimation + elimination saves time

Fluency in high-frequency routines can save a great deal of time:

  • Average speed for a round trip =2v1v2v1+v2=\frac{2v_1v_2}{v_1+v_2} (not the arithmetic mean!)
  • Successive discounts/growth: multiply, do not add
  • Proportional distribution: first find the total number of parts, then take by parts
  • Joint work rates: add the rates t=11/t1+1/t2t=\frac1{1/t_1+1/t_2}
  • Percentage change =new−oldold×100=\frac{\text{new}-\text{old}}{\text{old}}\times100

Key points

  • ◆Remember the formula for average speed on a round trip
  • ◆Discounts and growth are "multiplicative", not "additive"
  • ◆For work-rate problems, first convert to rates and then add them

Some questions give a solved case and ask you to apply the same structure to new numbers; or they test spatial reasoning (painted cube faces, path counting). Painted cube: after a cube with all faces painted n3n^3 is cut up, =8=8 have three painted faces, =12(n−2)=12(n-2) have two, =6(n−2)2=6(n-2)^2 have one, and =(n−2)3=(n-2)^3 have none.

Key points

  • ◆Map the new problem onto an already-solved structure
  • ◆Cube: 8 / 12(n-2) / 6(n-2)² / (n-2)³
  • ◆The sum of the four categories should equal n³

Worked examples

Try each example yourself before checking the explanation.

1Average speed for a round trip

Outbound journey 30 km/h, return along the same route 60 km/h, average speed for the whole journey?

2Successive discounts

£200 is first reduced by 10%, then reduced by 20% at this price; what is the final price?

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