EQN-03-14

Applied Word Problems: Money, Mixture & Distance

Set up equations for the three classic applied scenarios the GAT-E reuses.

~6 min read•Difficulty 3/5

Track the VALUE, not just the count. If there are n coins of 5 and $(12 - n)$ of 10:

value = 5 n + 10 (12 - n)

Set the value equal to the given total and solve.

Track the amount of the key ingredient before and after.

x litres of 30% acid + y litres of 50% acid \to 0.30 x + 0.50 y of pure acid

The pure-ingredient amounts add up; so do the volumes.

distance $=$ speed $\times$ time. Build one equation per traveller, then relate them.

Two cars leave the same point . One at 60 km / h, one at 80 km / h .

After t hours the gap is (80 - 60) t = 20 t .

Worked examples

Example 1 (difficulty 3/5)

A purse has 15 coins, all 5s and 10s, worth 110 in total. How many 10-coins are there?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

Two trains leave a station in opposite directions at 50 km/h and 70 km/h. After how many hours are they 360 km apart?

◎ Predict the answer first:

For coins/notes, write the total VALUE (count $\times$ denomination), not just counts.
For mixtures, the pure-ingredient amounts add and the volumes add.
Opposite directions add speeds; same direction subtracts them.

Summing coin counts instead of their values

Why: Counts are the obvious quantity.

Fix: Multiply each count by its denomination to get value.

Adding speeds for two travellers going the same way

Why: Adding feels like "combining".

Fix: Same direction

\to

subtract speeds (relative speed); opposite

\to

add.