Forming & Solving Equations (AQA GCSE Further Maths): Revision Note

Algebraic ratio & percentages

How do I change ratios and percentages into algebra?

• If $x:y$ equals $a:b$ then $\frac{x}{y}=\frac{a}{b}$

  • the equals sign now allows you to use algebra

• $x\%$ of $y$can be written $\frac{x}{100}\times y$
  

  • this simplifies to $\frac{xy}{100}$

• An increase of $a\%$ on $b$ can be written $\left(1+\frac{a}{100}\right)\times b$

  • this can also be written $b\left(1+\frac{a}{100}\right)$

Forming & solving equations

How do I form an expression or equation?

• An expression is an algebraic statement without an equals sign e.g. $3x+7$ or $2(x^2-14)$

• Sometimes we need to form expressions to help us express unknown values

• If a value is unknown you can represent it by a letter such as $x$

• An equation is simply an expression with an equals sign that can then be solved

• For example

  • If Adam is 10 years younger than Barry and the sum of their ages is 25, you can find out how old each one is

    • Represent Adam's age as $x$ then Barry's age is $x+10$

    • We can solve the equation $x+x+10=25$ or $2x+10=25$

• Sometimes you might have two unrelated unknown values (x and y) and have to use the given information to form two simultaneous equations

Many questions involve having to form and solve equations from information given about things relating to shapes, like lengths or angles.

How do I form an equation involving a 2D shape?

• If no diagram is given it is almost always a good idea to quickly sketch one

• Add any information given in the question to the diagram

  • This information will normally involve expressions in terms of one or two variables

• If the question involves perimeter, figure out which sides are equal length

  • If a triangle is given, are any of the sides equal length?

• If the question involves area, write down the necessary formula for the area of that shape

  • If it is an uncommon shape you may need to split it up into two or more common shapes that you can work out areas for

  • this is often called compound area in GCSE Mathematics courses

• Remember that a regular polygon means all the sides are equal length

  • For example, a regular pentagon with side length 2x – 1 has 5 equal sides so its perimeter is 5(2x – 1)

• If one of the shapes is a circle or part of a circle, use π throughout rather than multiplying by it and ending up with long decimals

• Consider the properties of angles within the given shape to decide which sides will have equal lengths

  • If a triangle is given, how many of the angles are equal?

    • An isosceles triangle has two equal angles

    • An equilateral triangle has three equal angles

  • Consider angles in parallel lines (alternative, corresponding, co-interior)

  • In a parallelogram or rhombus, opposite angles are equal and all four sum to 360°

  • A kite has one equal pair of opposite angles

• If the question involves angles, use the formula for the sum of the interior angles of a polygon

  • For a polygon of n sides, the sum of the angles will be 180°×(n - 2)

  • Remember that a regular polygon means all the angles are equal

• You may also have to use circle theorems to spot which angles are equal to each other, or to spot right-angles

How do I form an equation involving the surface area or volume of a 3D shape?

• If no diagram is given it is almost always a good idea to quickly sketch one

• Add any information given in the question to the diagram

  • This information will normally involve expressions in terms of one or two variables

• Consider the properties of the given shape to decide which sides will have equal lengths

  • In a cube all sides are equal

  • All prisms have the same shape (cross section) at the front and back

• If the question involves volume, write down the necessary formula for the volume of that shape

  • If it is an uncommon shape the exam question will give you the formula that you need

• It the question involves surface area,

  • Remember to consider any faces that may be hidden in the diagram

    • STEP 1
      Write down the number of faces the shape has and if any are the same

    • STEP 2
      Identify the 2D shape of each face and write down the formula for the area of each one

    • STEP 3
      Substitute the given expressions into the formula for each one

    • STEP 4
      Add the expressions together, double checking that you have one for each of the faces