1.1 Real Numbers, Ratios, Rates & Percentages

Real Numbers, Ratios, Rates & Percentages

The Mathematics subtest of the Ateneo College Entrance Test (ACET) demands extraordinary mental agility. Because calculators are strictly prohibited, your success depends on rapid number sense, quick identification of structural patterns, and instant execution of arithmetic shortcuts. The ACET tests real numbers not merely through straightforward computations, but through multi-step word problems involving rates, proportions, and percentage changes engineered to trigger common psychological traps.

---

1. Real Number Structure & Order of Operations

The real number system $\mathbb{R}$ consists of rational numbers $\mathbb{Q}$ (integers, terminating decimals, and repeating decimals) and irrational numbers $\mathbb{I}$ (non-repeating, non-terminating values such as $\sqrt{2}, \sqrt{3}, \pi,$ and $e$).

Order of Operations (PEMDAS / BODMAS)

When evaluating complex numerical expressions, strictly adhere to the established hierarchy:

1. Parentheses / Grouping: Evaluate operations inside brackets (), braces {} or fraction bars first.
2. Exponents / Radicals: Compute powers and roots from left to right.
3. Multiplication & Division: Perform these with equal priority strictly from left to right.
4. Addition & Subtraction: Perform these with equal priority strictly from left to right.

ACET Speed Tactic (Left-to-Right Priority): In the expression $24 \div 4 \times 2$, many test-takers improperly multiply $4 \times 2 = 8$ first to get $24 \div 8 = 3$. The correct execution is left-to-right: $(24 \div 4) \times 2 = 6 \times 2 = 12$.

Essential Divisibility Rules for Fast Prime Factoring

Quickly testing for divisibility allows you to reduce fractions and simplify radical terms instantaneously without long division:

---

2. High-Frequency Mental Math Conversions

Memorize these fraction, decimal, and percentage equivalents. Converting complex percentage problems into simple fractional multiplication is often the single fastest way to arrive at an answer.

$$\begin{aligned} \frac{1}{8} &= 0.125 = 12.5\% & \frac{3}{8} &= 0.375 = 37.5\% & \frac{5}{8} &= 0.625 = 62.5\% & \frac{7}{8} &= 0.875 = 87.5\% \\ \frac{1}{6} &\approx 0.1667 = 16.67\% & \frac{5}{6} &\approx 0.8333 = 83.33\% & \frac{1}{12} &\approx 0.0833 = 8.33\% & \frac{1}{16} &= 0.0625 = 6.25\% \\ \frac{1}{7} &\approx 0.142857 & \frac{2}{7} &\approx 0.285714 & \frac{1}{9} &= 0.\bar{1} = 11.\bar{1}\% & \frac{1}{11} &= 0.\overline{09} = 9.\overline{09}\% \end{aligned}$$

---

3. Ratios, Proportions & Partitive Partitioning

A ratio compares two quantities $a$ and $b$ as $a:b$ or $\frac{a}{b}$. A proportion states that two ratios are equivalent: $\frac{a}{b} = \frac{c}{d} \implies ad = bc$.

Part-to-Part vs. Part-to-Whole Relationships

If a mixture contains red and blue marbles in the ratio $3:7$:

• The part-to-part ratio of red to blue is $3:7$.
• The total number of units is $3 + 7 = 10$.
• The part-to-whole ratio for red marbles is $\frac{3}{10} = 30\%$, and for blue marbles is $\frac{7}{10} = 70\%$.

The Multiplier ($x$) Method for Partitive Proportions

When a total quantity $T$ is divided into a multi-part ratio $a : b : c$:

1. Express the parts as $ax$, $bx$, and $cx$.
2. Set up the sum: $ax + bx + cx = T \implies x(a + b + c) = T$.
3. Solve for $x = \frac{T}{a+b+c}$ and multiply by the required ratio coefficient.

---

4. Rates: Motion, Combined Work & Pipes

Rate of Motion & The Average Speed Trap

Distance formula: $d = r \cdot t \implies r = \frac{d}{t}$.

CRITICAL ACET TRAP (Average Speed): If a car travels from Town A to Town B at speed $v_1$ and returns along the exact same distance at speed $v_2$, the average speed is NOT the arithmetic mean $\frac{v_1 + v_2}{2}$. Because time spent on each leg varies, the overall average speed is the harmonic mean:

$$v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2 d}{\frac{d}{v_1} + \frac{d}{v_2}} = \frac{2 v_1 v_2}{v_1 + v_2}$$

Combined Work Rate Model

If Worker A completes a job in $t_A$ hours and Worker B completes the same job in $t_B$ hours, their individual rates are $R_A = \frac{1}{t_A}$ and $R_B = \frac{1}{t_B}$ jobs/hour. Working together, their combined rate $R_{\text{combined}}$ is:

$$R_{\text{combined}} = \frac{1}{t_A} + \frac{1}{t_B} = \frac{t_A + t_B}{t_A t_B} \implies T_{\text{together}} = \frac{t_A t_B}{t_A + t_B}$$

If Pipe A fills a tank in $F$ hours and Pipe B drains the tank in $D$ hours (where $D > F$), the net filling rate is:

$$R_{\text{net}} = \frac{1}{F} - \frac{1}{D} = \frac{D - F}{F D}$$

---

5. Variations & Proportional Dependencies

1. Direct Variation: $y \propto x \implies y = k x \implies \frac{y_1}{x_1} = \frac{y_2}{x_2}$
2. Inverse Variation: $y \propto \frac{1}{x} \implies y = \frac{k}{x} \implies x_1 y_1 = x_2 y_2$
3. Joint & Combined Variation: $y \propto \frac{x z}{w} \implies y = \frac{k x z}{w} \implies \frac{w_1 y_1}{x_1 z_1} = \frac{w_2 y_2}{x_2 z_2}$

---

6. Percentages, Percent Change & Successive Discounts

Basic Percent Translation Formula

Translate English statements directly into algebraic equations:

• "Is" $\rightarrow =$
• "Of" $\rightarrow \times$
• "What number / percent" $\rightarrow x$

$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$$

The Successive Discount / Increase Trap

Applying a discount of $d_1\%$ followed by another discount of $d_2\%$ is NOT equal to a single discount of $(d_1 + d_2)\%$.

• Final Price $= P_0 \times (1 - d_1) \times (1 - d_2)$
• Single Equivalent Discount Rate $= 1 - (1 - d_1)(1 - d_2)$

---

Worked Step-by-Step ACET Exam Problems

Problem 1: Combined Work with Variable Timelines

Worker A can build a wall in 12 hours. Worker B can build the same wall in 15 hours. Worker A starts working alone for 3 hours, after which Worker B joins to complete the rest. How many total hours were required to complete the wall?

Step 1: Determine individual work rates.

• Rate of A: $R_A = \frac{1}{12}$ wall/hour.
• Rate of B: $R_B = \frac{1}{15}$ wall/hour.

Step 2: Calculate work completed by A in the first 3 hours.

$$W_1 = R_A \times 3 = \frac{1}{12} \times 3 = \frac{1}{4} \text{ of the wall}$$

Step 3: Calculate remaining work.

$$W_{\text{remaining}} = 1 - \frac{1}{4} = \frac{3}{4} \text{ of the wall}$$

Step 4: Find combined rate of A and B.

$$R_{\text{combined}} = \frac{1}{12} + \frac{1}{15} = \frac{5 + 4}{60} = \frac{9}{60} = \frac{3}{20} \text{ wall/hour}$$

Step 5: Compute time required for remaining work.

$$t_{\text{together}} = \frac{W_{\text{remaining}}}{R_{\text{combined}}} = \frac{3/4}{3/20} = \frac{3}{4} \times \frac{20}{3} = 5 \text{ hours}$$

Step 6: Total time elapsed.

$$\text{Total Time} = 3 \text{ hours (alone)} + 5 \text{ hours (together)} = 8 \text{ hours}$$

---

Problem 2: Multi-Stage Successive Percent Change

An item's wholesale price is increased by $25\%$. During a weekend clearance sale, the new price is discounted by $20\%$. Later, an administrative surcharge of $10\%$ is added to the sale price. What is the net overall percentage change relative to the initial wholesale price?

Step 1: Set a convenient initial value. Let original wholesale price $P_0 = 100$.

Step 2: Apply stage 1 ($25\%$ markup).

$$P_1 = 100 \times (1 + 0.25) = 125$$

Step 3: Apply stage 2 ($20\%$ discount).

$$P_2 = 125 \times (1 - 0.20) = 125 \times 0.80 = 100$$

Step 4: Apply stage 3 ($10\%$ surcharge).

$$P_3 = 100 \times (1 + 0.10) = 110$$

Step 5: Compute net overall percentage change.

$$\text{Net Change} = \frac{110 - 100}{100} \times 100\% = +10\%$$

---

ACET Speed Tactics & Traps Summary