Grade A Direct and inverse proportions Write and

Grade A Direct and inverse proportions Write and interpret equations that describe direct and inverse proportion

Key Vocabulary Inverse Direct Proportional Equation

How to form directly proportional equations y is directly proportional to x. y = 20 when x = 4. By writing an equation of proportionality, work out the value of y when x = 7 The first sentence tells us that y ∝ x so this means that y = kx where k is a constant. Substitute y = 20 and x = 4 into the equation y = kx to find value of k. 20 = k × 4 This is our 5=k equation of proportionality So y = 5 x Now substitute x = 7 into our equation of proportionality y = 5 × 7 = 35 y = 35

How to form inversely proportional equations y is inversely proportional to x 2. When y is 2, x is 6 By writing an equation of proportionality, work out the value of y when x = 10 y is inversely proportional to x 2 tells us that y ∝ 1/x 2 this means that y = k/x 2 where k is a constant Substitute y = 2 and x = 6 into the equation y = k/x 2 to find value of k. 2 = k/62 This is our 2 = k/36 equation of 72 = k proportionality So y = 72/x 2 Now we can substitute x = 10 into our equation of proportionality and solve for y y = 72/102 y = 72/100 y = 18/25

Now you try… y is directly proportional to x 2. When y is 72, x is 6 By writing an equation of proportionality, work out the value of y when x = 5 y = 50 y is inversely proportional to x 4. When y is 8, x is 2 By writing an equation of proportionality, work out the value of y when x = 4 y = 1/2

Problem solving and reasoning The attractive force between 2 objects is inversely proportional to the square of their distance apart. The force is 0. 96 Newtons when the distance is 10 miles. What will the force be when the distance is 12 miles? Force ∝ 1/Distance 2 F = k/D 2 Substitute F = 0. 96 and D = 10 into F = k/D 2 0. 96 = k/102 0. 96 × 100 = k 96 = k F = 96/D 2 Substitute D = 12 into the equation above F = 96/122 F = 96/144 F = 2/3

Problem solving and reasoning The distance it is possible to see on a clear day, d miles, varies directly with the square root of the height h metres somebody is stood above sea level. Standing on a pier 9 metres above sea level it is possible to see 21 miles. Standing on top of a cliff it is possible to see the coast 36 miles away. How high are the cliffs to the nearest metre? Distance ∝ √Height d = k √h Substitute d = 21 and h = 9 21 = k √ 9 21 = 3 k 9=k d = 9 √h Substitute d = 36 36 = 9 √h 4 = √h 16 = h

Reason and explain True or false? y is directly proportional to the square of 2 x This means that y = 2 kx 2? y is inversely proportional to the square root of 4 x This means that y = k/(2√x)? y is directly proportional to the cube of x 2 This means that y = kx 6? y is inversely proportional to the third root of 27 x This means that y = 1/(3√x)?