1.1 Real Numbers, Ratios, Rates & Percentages
Key Takeaways
- Master mental math shortcuts and fraction-decimal-percent equivalencies to eliminate scratchwork delays under ACET time constraints.
- Apply the unit ratio method for partitive proportions and distinguish clearly between part-to-part and part-to-whole relationships.
- Solve combined work and average speed problems using harmonic mean concepts rather than simple arithmetic averages.
- Differentiate direct, inverse, and joint variation setups by establishing explicit constants of proportionality.
Last updated: July 2026
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:
- Parentheses / Grouping: Evaluate operations inside brackets
(), braces{}or fraction bars first. - Exponents / Radicals: Compute powers and roots from left to right.
- Multiplication & Division: Perform these with equal priority strictly from left to right.
- 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:
| Divisor | Divisibility Condition | Example | ACET Application |
|---|---|---|---|
| 3 | Sum of digits is divisible by 3 | 4,251 $\rightarrow 4+2+5+1 = 12$ (Divisible) | Fast fraction simplification |
| 4 | Last two digits form a number divisible by 4 | 7,316 $\rightarrow 16$ (Divisible) | Quick reduction of denominators |
| 6 | Number is even AND divisible by 3 | 8,532 $\rightarrow$ Even and $8+5+3+2=18$ | Factoring large composite values |
| 8 | Last three digits form a number divisible by 8 | 19,120 $\rightarrow 120 \div 8 = 15$ | Reducing higher-order ratios |
| 9 | Sum of digits is divisible by 9 | 6,858 $\rightarrow 6+8+5+8=27$ (Divisible) | Simplifying radical radicands |
| 11 | Alternating sum of digits is divisible by 11 | 9,185 $\rightarrow (9+8)-(1+5) = 11$ | Factoring special polynomials |
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.
\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 | Problem Type | Common Student Trap | ACET Speed Tactic | | :--- | :--- | :--- | | **Average Speed** | Averaging rates: $\frac{v_1+v_2}{2}$ | Use harmonic mean: $\frac{2v_1v_2}{v_1+v_2}$ when distance is equal | | **Combined Work** | Adding times directly: $t_1 + t_2$ | Add rates: $R_{\text{total}} = \frac{1}{t_1} + \frac{1}{t_2}$ | | **Successive Percent** | Adding percentages: $+20\% - 20\% = 0\%$ | Use multiplier method: $1.20 \times 0.80 = 0.96$ (Net $-4\%$) | | **Inverse Variation** | Setting up direct ratio: $\frac{x_1}{y_1} = \frac{x_2}{y_2}$ | Product is constant: $x_1 y_1 = x_2 y_2$ |Test Your Knowledge
A motorist drives from Manila to Baguio at an average speed of 60 km/h and returns along the exact same route at an average speed of 40 km/h. What is the driver's overall average speed for the entire round trip?
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A sum of 12,000 pesos is divided among Alice, Bob, and Charlie in the ratio 2 : 3 : 5. What is the difference between Charlie's share and Alice's share?
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If 8 identical water pumps can drain a flooded reservoir in 15 hours, how many hours would it take 12 of these same pumps to drain the reservoir?
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