TARA ratio questions: no-calculator methods

Three partners put in £4,000, £6,000 and £10,000 and share a profit of £5,400 in the same ratio. How much does the partner who put in £6,000 receive?

Ratio $2 : 3 : 5$, so 10 parts. One part is $5400 \div 10$ $= 540$. Three parts: £1,620.

Why this works: Simplify the contributions to 2:3:5, then three parts of £540.

Why the tempting route fails: Splitting equally gives £1,800 each: the question says the profit follows what each put in.

A bag holds red and blue counters in the ratio 3:5. After 12 red counters are added, there are equal numbers of red and blue. How many counters were in the bag at first?

Red $3k$, blue $5k$. After adding: $3k + 12$ $= 5k$, so $k$ $= 6$. At first: $8k$ $= 48$ counters.

Why this works: The blue count does not change, so it anchors the equation: 48 counters.

Why the tempting route fails: 60 is the number after the 12 are added. Read which moment the question asks about.