Question: Simplify the ratios (a) $240$ g $: 1.8$ kg and (b) $0.36 : 1.2$.

Step 1. A ratio can only be simplified when both parts are in the same unit. Convert first:

$$240 \text{ g} : 1.8 \text{ kg} = 240 \text{ g} : 1800 \text{ g}$$

Step 2. Drop the units and divide both sides by the HCF ($120$):

$$240 : 1800 = 2 : 15$$

Step 3. For decimals, multiply both parts up to whole numbers first ($\times 100$), then simplify:

$$0.36 : 1.2 = 36 : 120 = 3 : 10$$

Step 4. Final form check: whole numbers, no units, no common factor left. Both answers pass.

A simplified ratio also reads as fractions of the whole: in $2 : 15$, the first quantity is $\dfrac{2}{17}$ of the total — the denominator is the sum of the parts.

The same clean-up order works every time: same units → whole numbers → divide by the HCF.

⚠ Watch out: $240 : 1.8$ simplified directly is nonsense — the unit mismatch silently scales one side by 1000. Convert to the smaller unit, then simplify; and a final ratio never carries units or decimals.

Question: Given $a : b = 4 : 9$ and $b : c = 21 : 16$, find $a : b : c$ in its simplest form.

Step 1. The two ratios share $b$ — but written as $9$ in one and $21$ in the other. Make the $b$'s equal by scaling both ratios to the LCM of $9$ and $21$, which is $63$.

Step 2. Scale the first ratio by $7$:

$$a : b = 4 : 9 = 28 : 63$$

Step 3. Scale the second ratio by $3$:

$$b : c = 21 : 16 = 63 : 48$$

Step 4. With $b$ matching, chain them:

$$a : b : c = 28 : 63 : 48$$

(No common factor divides all three, so this is simplest form.)

Check. $a : b = 28 : 63 = 4 : 9$ ✓ and $b : c = 63 : 48 = 21 : 16$ ✓ — the combined ratio must reproduce both originals.

Once combined, the triple ratio unlocks sharing questions: 28 + 63 + 48 = 139 parts of any given total.

⚠ Watch out: The shared quantity must be represented by the same number before joining — gluing $4 : 9$ onto $21 : 16$ directly (giving $4 : 9 : 16$ or similar) is the standard error. Scale to the LCM of the two $b$-values, and scale every part of each ratio, not just the $b$.

Question: Ali and Ben have money in the ratio $5 : 3$. After Ali spends $\usd{}20$ and Ben receives $\usd{}20$, the ratio becomes $7 : 5$. How much did each have at first?

Step 1. Turn the original ratio into algebra with one unit:

$$\text{Ali} = 5u, \qquad \text{Ben} = 3u$$

Step 2. Write the after amounts and equate to the new ratio:

$$\dfrac{5u - 20}{3u + 20} = \dfrac{7}{5}$$

Step 3. Cross-multiply and solve:

$$5(5u - 20) = 7(3u + 20)$$ $$25u - 100 = 21u + 140 \;\Rightarrow\; 4u = 240 \;\Rightarrow\; u = 60$$

Step 4. Answer the actual question — the original amounts:

$$\text{Ali} = 5(60) = \usd{}300, \qquad \text{Ben} = 3(60) = \usd{}180$$

Check. After: $300 - 20 = 280$ and $180 + 20 = 200$; $\;280 : 200 = 7 : 5$. ✓

⚠ Watch out: The two ratios use different units — $5:3$ and $7:5$ describe different totals, so never mix their parts in one expression; the single unknown $u$ belongs to the original ratio only. And read who gains and who spends carefully: swapping the $\pm 20$ produces a tidy-looking wrong answer.