Units of Measurements

So $2.4\text{ km} = 2.4 \times 1000 = 2400\text{ m}$, and $850\text{ g} = 850 \div 1000 = 0.85\text{ kg}$.

This year the conversions get trickier in two ways: area and volume units use different conversion numbers, and you handle the accuracy of measurements much more carefully. Both build on this same multiply-or-divide thinking.

• Length, mass and capacity convert by powers of 10.
• Smaller unit → bigger number → multiply.
• Bigger unit → smaller number → divide.
• Area and volume units use different conversion numbers.

So the area rule is to square the length conversion factor:

• $1\text{ cm}^2 = 10 \times 10 = 100\text{ mm}^2$
• $1\text{ m}^2 = 100 \times 100 = 10\,000\text{ cm}^2$

Volume units go a step further — you cube the factor, because volume has three dimensions:

• $1\text{ cm}^3 = 10 \times 10 \times 10 = 1000\text{ mm}^3$
• $1\text{ m}^3 = 100 \times 100 \times 100 = 1\,000\,000\text{ cm}^3$

A handy capacity link: $1\text{ cm}^3$ of space holds exactly $1\text{ ml}$, so $1000\text{ cm}^3 = 1\text{ litre}$. When a volume problem mixes units, convert everything to the same unit first.

• Area units: square the length conversion factor.
• $1\text{ m}^2 = 10\,000\text{ cm}^2$, not 100.
• Volume units: cube the length conversion factor.
• $1\text{ cm}^3 = 1\text{ ml}$, so $1000\text{ cm}^3 = 1\text{ litre}$.

The triangle trick works for all three. Cover the quantity you want and read off the rest. For pressure: cover F and you see $\text{P} \times \text{A}$; cover A and you see $\text{F} \div \text{P}$.

So a force of $48\text{ N}$ spread over an area of $6\text{ cm}^2$ gives a pressure of $48 \div 6 = 8\text{ N/cm}^2$. A block of mass $90\text{ g}$ and volume $30\text{ cm}^3$ has density $90 \div 30 = 3\text{ g/cm}^3$.

• A compound unit combines two measures, like km/h, g/cm³ or N/cm².
• Speed = distance ÷ time; density = mass ÷ volume; pressure = force ÷ area.
• A triangle helps you rearrange any of these relationships.
• The word 'per' in a unit always means 'divided by'.

Density and pressure rearrange the same way. To find a mass from a density, use $\text{mass} = \text{density} \times \text{volume}$. To find an area from a pressure, use $\text{area} = \text{force} \div \text{pressure}$. The triangle is the same idea every time.

• Write down what you have and what you want before calculating.
• Use the triangle to choose multiply or divide.
• Convert units to match before substituting — for example minutes to hours.
• Mass = density × volume; area = force ÷ pressure.

The "half a unit" depends on the rounding. To the nearest $10$, a mass of $40\text{ kg}$ has bounds of $35\text{ kg}$ and $45\text{ kg}$. To one decimal place, a time of $3.2\text{ s}$ has bounds of $3.15\text{ s}$ and $3.25\text{ s}$.

• A rounded measurement gives a range of possible true values.
• Lower bound = rounded value − half a unit.
• Upper bound = rounded value + half a unit.
• The 'half a unit' depends on the degree of accuracy used.

Step-by-step solution

1. Step 1

   Density is mass divided by volume.

   $\text{density} = \frac{\text{mass}}{\text{volume}}$

2. Step 2

   Substitute the mass and the volume.

   $\text{density} = \frac{90}{30} = 3$