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This page sets out the method for ratio as a fraction or linear function in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Ratio as a fraction or linear function topic page.
When you see 'what fraction of the total is …' and a ratio is given, add the parts first.
When you see 'write a formula connecting and ' or '', build .
Replace x, multiply by the gradient, then add the intercept.
y = (3/4)x when x = 8 is 6, because 3/4 of 8 is 6.
A machine that multiplies by 2 and then adds 3 turns 4 into 11.
If y is known, undo the intercept and then divide by the gradient.
Multiply before you add, unless the algebra already shows brackets.
The number after “when x =” is the only input.
Red to blue in the ratio 3:5 means 3 red parts and 5 blue parts. There are 8 parts in the whole. Red is 3/8 of the counters, not 3/5. The fraction 3/5 would compare red with blue, which is the ratio written another way, not red’s share of the total.
If there are 48 counters, red is 3/8 × 48. Divide 48 by 8 to get 6 counters in one part, then multiply by 3. The same one-part idea used for sharing is this fraction in disguise. The fraction makes the “of the whole” language precise.
Method: 1. Add the ratio numbers to get the denominator. 2. Put the part you want in the numerator. 3. Multiply that fraction by the whole amount. 4. Check that the two fractions of the whole add to 1.
Substitute means replace x with the number you were given, then follow the order of operations. In y = 4x + 7, the 4 and the x are joined by multiplication. That product happens before the +7. For x = 6 the line is 4 × 6 + 7.
The intercept is not optional. Leaving it off gives the product only, which is a point on a different line. Adding the intercept before multiplying, as in 4 × (6 + 7), invents brackets the rule does not have. Write the substitution, then the product, then the sum.
Method: 1. Replace x with the given number. 2. Multiply by the gradient. 3. Add the intercept afterwards. 4. Read the result as the value of y.
A rule such as y = 3/4x scales x by three quarters. You may divide by 4 first and then multiply by 3, or multiply by 3 and then divide by 4. Both orders agree when the arithmetic is exact. For a whole-number x that is a multiple of 4, dividing first keeps the numbers small.
The numerator is part of the scale. Dividing by 4 and forgetting to multiply by 3 finds one quarter, not three quarters. If the fraction is greater than 1, such as 5/2, the output is larger than the input. That is still a multiplier, not an error.
Method: 1. Identify the numerator and the denominator. 2. Divide x by the denominator, or multiply by the numerator first. 3. Complete the other part of the fraction. 4. Check that three quarters is more than a half and less than the whole.
If y is known and x is not, undo the intercept before you undo the gradient. The machine added last, so subtract first. For y = 3/4x + 5 and y = 17, subtract 5 to get 12. That 12 is 3/4 of x.
Undoing a fraction means multiplying by its reciprocal. 3/4 of x is 12, so x = 12 × 4/3 = 16. Check by running forwards: three quarters of 16 is 12, and 12 + 5 = 17. A check that returns the given y shows the inverse steps were in the right order.
Method: 1. Subtract the intercept from y. 2. The remaining term is the gradient multiplied by x. 3. Divide by the gradient, or multiply by its reciprocal. 4. Substitute your x back into the rule and check y.
A number machine is a rule written as a chain of operations, such as input, × 4, -5, output. To find an output, work from left to right and do each operation in turn. To find an input from a known output, run the machine backwards. Start at the output and undo the last operation first, using the inverse: undo -5 by adding 5, then undo × 4 by dividing by 4.
This is the same as solving an equation. If the machine is × 4 then -5 and the output is 43, then 4x - 5 = 43, so 4x = 48 and x = 12. Both ways give the same answer. In exams, a number machine usually earns one mark for the forward step and two marks for the backward step. Show each undo step on its own line so that a slip in the first undo does not lose every mark.
Method: 1. To find an output, apply each operation in order, left to right. 2. To find an input, start with the output and use the inverse of each step, in reverse order. 3. Write the number after each step. 4. Put the answer back into the machine to check that it gives the output.
Tip: Backwards means reverse order and opposite operations. Both parts matter.
Many exam formulas describe a cost: a fixed charge plus a charge for each unit. A plumber who charges £35 to call out and £20 for each hour has the formula C = 35 + 20h, where C is the total cost in pounds and h is the number of hours. The 35 is the intercept and 20 is the gradient, or the rate per hour. The units are the units of the letters, so say them in the answer.
To use the formula, substitute and follow the order of operations: multiply first, then add. For h = 3.5, work out 20 × 3.5 = 70, then add 35 to get 105. To go backwards, set the formula equal to the known total and solve. From 175 = 35 + 20h, subtract 35 to get 140 = 20h, then divide by 20 to get h = 7. Write the answer with its unit.
Method: 1. Identify the fixed amount and the amount per unit. 2. Write the formula as fixed charge plus rate times units. 3. To find a total, substitute and multiply before adding. 4. To find the number of units, subtract the fixed amount, then divide by the rate.
A ratio such as red : green : blue = 2 : 3 : 5 can be turned into fractions of the total. There are 10 parts, so green is 3/10 of the total T. As a rule, green = 3/10T. This is a linear function with no intercept. When T = 150, green is 3/10 × 150 = 45.
Going backwards is the same idea as running a machine backwards. If 45 pens are green, then 3/10T = 45. Multiply by 10 and divide by 3 to get T = 150. A popular slip is to multiply by 5/3 or use the wrong part of the ratio. The total always involves the sum of all the parts in the denominator, so check that you have added every number in the ratio. A quick test is to confirm that all the fractions of the whole add up to one.
Method: 1. Add all the parts of the ratio to get the number of parts. 2. Write the colour you need as a fraction of the total. 3. Multiply the total by that fraction to go forwards. 4. To go backwards, multiply by the reciprocal of the fraction.
A rule y = mx + c can be found from any two input and output pairs. First work out the gradient: the change in output over the change in input. If x = 2 gives 11 and x = 6 gives 23, then m = (23 - 11)/(6 - 2) = 3. Then substitute one pair to find c: 11 = 3 × 2 + c, so c = 5. The rule is y = 3x + 5.
A "show that" question gives a target, often that something is less than a percentage of an actual value. Work out the estimate from the formula, find the difference with the real value, and then find the percentage of the real value. Compare the two numbers and write a conclusion in words. The marks come from each of these steps, not only from the final comparison.
Method: 1. Find the gradient: change in output divided by change in input. 2. Substitute one pair into y = mx + c to find c. 3. Check the rule with the second pair. 4. For "show that", work out each quantity, compare, and write a closing sentence.
Tip: If the rule gives the wrong value for the second pair, the gradient is wrong. Check the subtraction.
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