Algebraic expressions · Ages 13–14
Formulae & algebraic fractions
From “Simplifying algebraic fractions” to “A new subject on both sides or below the line”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- Simplifying algebraic fractionsFactorise first; cancel factors, never terms
Method3x + 6x2 + 2x = 3x + 2x(x + 2) = 3x
Another way: Check with a numberx = 2: 128 = 32, 3x = 32
- Multiplying and dividing algebraic fractionsCancel, then multiply; to divide, multiply by the reciprocal
Method4x + 1 × x + 12x = 2x
Another way: Multiply, then simplify4(x + 1)2x(x + 1) = 2x
- Adding and subtracting algebraic fractionsCommon denominator first; bracket a top that is taken away
Method2x + 1 + 3x − 2 = 5x − 1(x + 1)(x − 2)
Another way: Check with a numberx = 3: 24 + 31 = 72, 144 × 1 = 72
- Substituting into formulaeSubstituted, it is an equation; bracket a negative value
Methodv = u + at: 30 = 6 + 4a, a = 6
Another way: Change the subject firsta = v − ut = 30 − 64 = 6
- Changing the subject of a formulaTreat the new subject as the unknown; do the same to both sides
Methodv = u + at: v − u = at, a = v − ut
Another way: Work backwardsUndo × t, + u: − u, then ÷ t
- A new subject on both sides or below the lineClear the denominator, gather it on one side, take it out
Methodax = b + cx: x(a − c) = b, x = ba − c
Another way: Check with numbersy = 3x − 2, x = 5 gives y = 1: 3 + 2 × 11 = 5
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1Simplifying algebraic fractions →
Simplify: 25x2y/(15xy2) = xy
Fill in each blank
Show the answer and working
Answer: 5, 3
Divide the top and the bottom by 5xy: x2y ÷ xy = x, xy2 ÷ xy = y, 25 ÷ 5 = 5, 15 ÷ 5 = 3.
Example 2Substituting into formulae →
In the formula v = u + at, u = 20, a = 3 and t = 8: v =
Show the answer and working
Answer: 44
v = 20 + 3 × 8 = 44. Times before plus.
Example 3A new subject on both sides or below the line →
Make x the subject of 15x + 1 = y: x = y −
Fill in each blank
Show the answer and working
Answer: 15, 1
Take 1 from both sides: 15x = y − 1; multiply by x, then divide by y − 1: x = 15y − 1.
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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