Topic 5

Scale Drawings, Mixtures, and Density Problems

This topic connects ratios to real-world proportional reasoning:

Scale drawings: represent real dimensions proportionally. Scale factor $=\dfrac{\text{drawing}}{\text{actual}}$ . Multiply or divide by the scale to convert between the two.
Mixtures: combine substances using ratio or weighted average relationships.
Density: expresses mass per unit volume:
$\text{density} = \frac{\text{mass}}{\text{volume}}.$
Rearrange as $m = dV$ or $V = \frac{m}{d}$ .

Core Skills

Use proportion equations for scale drawings: $\dfrac{\text{drawing}}{\text{actual}}=\dfrac{x_1}{x_2}$ .
For mixtures, set up equations based on total quantity and total content (e.g. salt, acid, cost).
For density, solve for missing variables given any two.
Keep track of units consistently across quantities.

On a map, 1 cm represents 5 km. If two towns are 7.2 cm apart on the map, how far apart are they in reality?

1:5 = 7.2:x \Rightarrow x = 7.2 \times 5 = \boxed{36\text{ km}}.

A model car is built at a scale of 1:20. The real car is 4 m long.

\text{model length} = \frac{1}{20}\times 4 = \boxed{0.2\text{ m}}.

A chemist mixes 3 L of 20% acid with 2 L of 50% acid.

\text{total acid} = 3(0.20)+2(0.50)=0.6+1.0=1.6\text{ L}.

Total volume

=5\text{ L}

\text{percent}=\frac{1.6}{5}\times100\%=32\%

. Final mixture: 32% acid.

A cube has a volume of

200\text{ cm}^3

and a density of

1.6\text{ g/cm}^3

. Mass

=dV=1.6(200)=\boxed{320\text{ g}}

A liquid with density

0.8\text{ g/cm}^3

has a mass of

400\text{ g}

V=\frac{m}{d}=\frac{400}{0.8}=500\text{ cm}^3.

Answer:

\boxed{500\text{ cm}^3}