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This page sets out the method for problem-solving with ratio & algebra 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 Problem-solving with ratio & algebra topic page.
When the ratio changes after something is added or removed, introduce for one part and build an equation from the new ratio.
When you have and and need , match the shared part.
If a pen costs £1 more than a pencil, replace each pen with a pencil plus £1.
Share £40 in the ratio 3:5. The larger share is 25.
If one person is twice as old as the other and together they are 36, the younger is 12.
2 adults and 3 children cost £28. A child costs £4. An adult costs £8.
Add the ratio parts before you split the amount.
Call the cheaper price, or the younger age, x. Write the dearer one from it.
Let the cheaper item be x pounds. If a pen costs £3 more than a pencil, a pen is x + 3. Two pens and four pencils are 2(x + 3) + 4x. That expression is the whole bill. Set it equal to £18 and solve. Expand the bracket before you collect like terms.
The £3 is added once for each pen, not once for the whole bill. Two pens contribute 2 × 3 = 6 extra pounds. Forgetting the bracket and writing 2x + 3 only charges the extra once. The equation then describes a different shop bill.
Method: 1. Let the cheaper price be x. 2. Write the dearer price as x plus the fixed extra. 3. Build the bill and set it equal to the total. 4. Expand, collect like terms, and divide.
Parts in the ratio 3:5 can be called 3x and 5x. If the parts are people and there are 40 pupils, then 3x + 5x = 40. The letter x is one part, the same idea as dividing the total by 8. Algebra keeps that idea in one line.
Solve for x, then multiply by the part you were asked for. Girls might be the 5x. Do not stop at x if the question asked for a group. Substituting back, 3x + 5x should equal 40. If it does not, the division slipped.
Method: 1. Let one part be x. 2. Write each group as its ratio number times x. 3. Add the groups and set the sum equal to the total. 4. Solve for x, then multiply to get the group you need.
If Sam is twice as old as his sister, the sister is x and Sam is 2x. Their ages add to 45, so x + 2x = 45. Three lots of the younger age make 45, so the younger age is 15 and Sam is 30. The 2 is inside Sam’s expression. It is not added after the total.
A later year changes both ages by the same amount. In four years, Tom’s age x becomes x + 4 and Leah’s age 2x becomes 2x + 4. The sum gains 8, not 4. Both people get older. Missing one of the +4 terms solves a different sentence.
Method: 1. Let the younger age be x. 2. Write the older age as 2x if it is twice as much. 3. Add the ages now, or add four years to each age if the sentence is about the future. 4. Solve, then check the ages against the original sentence.
If a child ticket is known, it is a number, not a second letter. Two adult tickets and three child tickets at £7 give the equation 2a + 3 × 7 = 39. Work out the children first. They cost £21. The adults share what is left.
Forming one equation means you do not need a second unknown. The known price is substituted before you solve. Divide the adults’ part of the bill by the number of adults, not by the number of tickets of both kinds. A check is the full bill rebuilt from your adult price and the given child price.
Method: 1. Let the unknown ticket price be a letter. 2. Substitute the known price. 3. Write one equation for the whole bill. 4. Solve, then rebuild the bill as a check.
Almost every problem in this topic starts with a letter for the smallest or simplest quantity. Choose it, then write each other quantity using the words. "40p more than a cookie" is c + 40. "Three times as old as Zoe" is 3x. "Five years older" is x + 5. State what the letter stands for, such as "let c be the cost of a cookie in pence", before you write anything else.
Then find the sentence in the question that gives a total or a link and turn it into an equation. "3 muffins and 2 cookies cost 420p" becomes 3(c + 40) + 2c = 420. Notice the bracket: the extra 40p applies to every muffin, so it has to be multiplied by 3. Write the units on the equation, solve, and put the unit in the final answer.
Method: 1. Write "let ... be ..." for the unknown, with its unit. 2. Write every other quantity in terms of that letter, using brackets for "each". 3. Form an equation from the sentence that gives the total. 4. Solve, then answer the question that was asked and check it.
Tip: Say what the question wants before you start. Finding x is often not the last step.
When money is shared in the ratio 5 : 3, write the shares as 5x and 3x. A total gives 5x + 3x = 8x. A difference gives 5x - 3x = 2x. If Ali gets 36 pounds more than Bo, then 2x = 36, x = 18, and the total is 8x = 144 pounds. The letter x is the value of one part, so always multiply back to get the share you were asked for.
The same method handles problems where one person gives some money to another so that they end up equal. The totals stay the same. If Rita and Sam share 96 sweets in the ratio 5 : 3, they have 60 and 36. For them to be equal each needs 48, so Rita gives 12. A quick check is that what Rita loses is what Sam gains.
Method: 1. Write the shares as multiples of x using the ratio. 2. Use the total or the difference to form an equation in x. 3. Solve for x and multiply to find each share. 4. Check that the shares add to the total and have the right difference.
A common harder question says two amounts are in the ratio 3 : 5, then the same amount is added or taken from one or both, and the new ratio is given. Write the amounts as 3k and 5k. After a change they become, for example, 3k - 8 and 5k. The new ratio 1 : 2 gives the equation (3k - 8)/5k = 1/2.
Cross-multiply to remove the fractions: 2(3k - 8) = 5k. Expand, collect the k terms, solve, then substitute to find the amounts. 6k - 16 = 5k gives k = 16, so the amounts are 48 and 80. Always test your answer by putting the changed amounts back into the new ratio. 40 : 80 simplifies to 1 : 2.
Method: 1. Write the starting amounts as multiples of k. 2. Apply the change to each amount. 3. Write the new ratio as a fraction equation and cross-multiply. 4. Solve for k, find the amounts, and check the new ratio.
If A : B = 3 : 4 and B : C = 6 : 5, make the shared term match. B is 4 in the first and 6 in the second, so scale both to 12: A : B = 9 : 12 and B : C = 12 : 10. So A : B : C = 9 : 12 : 10, which is 31 parts. Dividing a total of 124 by 31 gives 4 per part. C gets 10 × 4 = 40.
Two different items with two different prices sometimes lead to two statements, such as "2 adult and 4 child tickets cost 42 pounds" and "1 adult and 5 child tickets cost 39 pounds". Halve the first to get a + 2c = 21. Subtract from the second: 3c = 18, so c = 6 and a = 9. Check in the original: 2 × 9 + 4 × 6 = 42.
Method: 1. Find a common value for the shared term and scale both ratios. 2. Join the ratios into one and add the parts. 3. For two unknowns, write two equations and eliminate one letter. 4. Substitute back into both statements to check.
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