Ratios and Rearranging Equations

In short: A ratio compares amounts, and you share in a ratio by adding the parts and finding the size of one part. To rearrange a formula, do the same thing to both sides: density $\rho =\frac{m}{V}$ gives $m=\rho V$ and $V=\frac{m}{\rho }$.

Ratios

A ratio compares two amounts. It is written with a colon: syrup : water = 1 : 4.

Simplify a ratio by dividing both sides by the same number, just like a fraction.

$$12:8=3:2(\text{divide both by }4)$$

Scale a ratio by multiplying both sides by the same number.

$$1:4\to 5:20(\text{multiply both by }5)$$

Sharing in a ratio

To split an amount in a ratio:

1. Add the parts.
2. Divide the total by the number of parts to find one part.
3. Multiply.

Percentages

$$\text{percentage}=\frac{\text{part}}{\text{whole}}\times 100\%$$

Proportion

Two quantities are directly proportional when doubling one doubles the other. If 2 g of magnesium makes 3.3 g of magnesium oxide, then 4 g makes 6.6 g.

The unitary method finds the amount for 1 first:

• 2 g makes 3.3 g, so 1 g makes $3.3÷2=1.65$ g.
• 5 g makes $5\times 1.65=8.25$ g.

Rearranging equations

Many chemistry formulas have three quantities. For example, density:

$$\rho =\frac{m}{V}$$

where $\rho$ (the Greek letter rho) is density, $m$ is mass and $V$ is volume.

To find a different quantity, rearrange. The golden rule: do the same thing to both sides.

To find $m$: multiply both sides by $V$.

$$m=\rho V$$

To find $V$: start with $m=\rho V$ and divide both sides by $\rho$.

$$V=\frac{m}{\rho }$$

Quick summary

• A ratio compares amounts. Simplify or scale by doing the same to both sides.
• To share in a ratio: add the parts, find one part, multiply.
• Percentage $=\frac{\text{part}}{\text{whole}}\times 100\%$.
• Rearranging: do the same thing to both sides, or use the formula triangle.
• Density: $\rho =\frac{m}{V}$, so $m=\rho V$ and $V=\frac{m}{\rho }$.
• Convert units before you substitute.

Quiz