Ratio Problems: Sharing, Changing and Combining
Solve ratio problems where a total is shared, one part is known, a ratio changes, or two ratios have to be joined through the quantity they share.
What a learner can do afterwards
- Find the total from one known part of a ratio
- Combine a : b and b : c into a : b : c through the shared term
- Work out the new ratio after a stated change to one part
1 · Read
Every ratio problem runs on one part. To share 50 pounds in ratio 3 to 2, add the parts to get 5, divide 50 by 5 to get 10 per part, and hand out 30 and 20. To find a total from one share, reverse the flow: 24 pounds covering 2 parts means 12 per part, and 5 parts make 60.
Combining ratios hinges on the shared term. With a to b as 2 to 3 and b to c as 4 to 5, b is 3 in one and 4 in the other. Scale both until b matches at 12, giving 8 to 12 to 15. Copying numbers across as 2 to 3 to 5 breaks the chain.
Changing a ratio means changing the amounts first. A 3 to 2 split of 50 pounds gives 30 and 20. Hand the first partner 10 more from outside to make 40 against 20. The new ratio 40 to 20 simplifies to 2 to 1.
Proportions solve ratio equations by cross-multiplying. If 2 parts match 14 litres of red paint, set 2 over 5 equal to 14 over w to find 35 litres of white. Read each ratio aloud first, since swapping the order on one side is the classic slip.
Find one part, match the shared term, and recompute amounts before simplifying.
More to read
5.4 Ratios and Proportions
Ratio problems come in three flavours that all run on one part. Sharing splits a total by the parts, like £50 in 3:2 giving £30 and £20. Finding a total from one share reverses the flow: £24 covering 2 parts means £12 per part and £60 altogether.
Changing a ratio means changing the amounts first. Handing the £30 partner £10 more makes £40 against £20, and the new ratio 40:20 simplifies to 2:1. Always recompute the amounts, then write and simplify the new ratio.
6.5 Solve Proportions and their Applications
Proportions solve ratio equations by cross-multiplying. If 2 parts correspond to 14 litres of red paint, 5 parts of white follow from 2/5 = 14/w, giving w = 35 litres. Setting the two ratios equal turns words into one equation.
Unit checks keep proportions honest. Both sides of 2/5 = 14/w compare red to white in the same order, litres to litres. Swapping the order on one side is the classic slip, so read each ratio aloud before crossing.
Section 10: Equivalent Ratios
Combining ratios hinges on the shared term. With a:b = 2:3 and b:c = 4:5, b is 3 in one and 4 in the other, so scale both until b matches at 12. The combined ratio 8:12:15 carries every link faithfully.
Equivalent ratios are the engine underneath. Multiplying all parts of a ratio by the same number keeps its meaning, which is why 8:12 still says a:b = 2:3. Copying numbers across without matching the shared term breaks the chain, as 2:3:5 wrongly claims.
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2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.