Chapter 2 — Ratios, proportions, and percentages

This chapter connects arithmetic and algebra through one central idea: comparing quantities. When we say that a mixture uses 2 parts water for 1 part concentrate, we are using a ratio. When two ratios express the same relationship, we have a proportion.

These ideas appear frequently in engineering contexts: drawing scales, energy use per hour, unit cost, and concentration.

2.1 Ratio between quantities

A ratio compares two compatible quantities. If a lab section has 18 students and 12 available computers, then the students-per-computer ratio is

\[ \frac{18}{12} = \frac{3}{2} = 1.5 \]

Interpretation: on average, there are 1.5 students per computer. If we invert the reading, we compute computers per student:

\[ \frac{12}{18} = \frac{2}{3} \]

The inversion changes the meaning, so it is essential to state clearly what is in the numerator and denominator.

2.2 Proportion and equivalent ratios

A proportion is an equality between two ratios:

\[ \frac{a}{b} = \frac{c}{d}, \quad b \neq 0,\ d \neq 0 \]

In this case, cross multiplication gives

\[ ad = bc \]

which is a direct test for equivalent ratios. For example,

\[ \frac{4}{10} = \frac{2}{5} \]

because

\[ 4 \cdot 5 = 10 \cdot 2 = 20. \]

2.3 Percentage as a base-100 ratio

A percentage is a standardized ratio:

\[ p\% = \frac{p}{100} \]

Quick conversions:

• \(25\% = 0.25\);
• \(8\% = 0.08\);
• \(135\% = 1.35\).

If a product costs $200 and receives a \(15\%\) discount, the discount value is

\[ 0.15 \cdot 200 = 30 \]

so the final price is $170.

Reference table

Situation	Expression	Result
10% of 80	\(0.10 \cdot 80\)	8
35% of 60	\(0.35 \cdot 60\)	21
12% increase on 50	\(50(1+0.12)\)	56
20% reduction on 90	\(90(1-0.20)\)	72

2.4 Rule of three and proportional reasoning

The simple rule of three organizes two-quantity proportional problems.

Worked example

A machine produces 120 parts in 3 hours at a constant rate. How many parts will it produce in 5 hours?

1. Write the proportion:

\[ \frac{120}{3} = \frac{x}{5} \]

1. Cross multiply:

\[ 120 \cdot 5 = 3x \]

1. Isolate \(x\):

\[ x = \frac{600}{3} = 200 \]

So the expected production in 5 hours is 200 parts.

2.5 Algebraic manipulation in proportional contexts

Algebra lets us solve proportions in a general way. Consider

\[ \frac{x}{12} = \frac{7}{8} \]

Multiply both sides by 12:

\[ x = 12\cdot\frac{7}{8} = \frac{84}{8} = 10.5 \]

A common percentage model is

\[ V_f = V_i(1+r) \]

• if \(r>0\), we have an increase;
• if \(r<0\), we have a decrease.

Example: with \(V_i=500\) and \(r=-0.08\),

\[ V_f = 500(1-0.08) = 500\cdot 0.92 = 460. \]

2.6 Matrix notation for repeated percentage updates

When several rates apply to several initial values, matrix notation organizes the computation. Let

\[ \mathbf{v}_i= \begin{bmatrix} 500\\ 300 \end{bmatrix}, \qquad R= \begin{bmatrix} -0.08 & 0\\ 0 & 0.12 \end{bmatrix} \]

Then one-step updated values can be written as

\[ \mathbf{v}_f = (I+R)\mathbf{v}_i \]

where \(I\) is the identity matrix. This chapter remains arithmetic-first, but this notation prepares the matrix language used later.

2.7 Chapter wrap-up

Ratios, proportions, and percentages form an essential block for quantitative modeling. In the rest of Volume 1, this reasoning supports data interpretation, algebraic problem solving, and the transition to function-based models.