7.3 Ratios, Proportions & Basic Algebra Without a Calculator

7.3 Ratios, Proportions & Basic Algebra Without a Calculator

FCTC's study guide lists ratios and algebra by name among the math topics, next to fractions, decimals, percentages, angles, area, and volume. Its sample set includes a classic ratio-split problem: a pass-to-fail ratio is given along with a class size, and you must find how many failed. These questions reward a short, repeatable method more than raw arithmetic speed.

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Ratios: Part-to-Part vs. Part-to-Whole

A ratio compares two quantities: 3 to 2, 3:2, or 3/2. Always check what the ratio is comparing:

• Part-to-part: "The ratio of engines to trucks is 3:2" compares two groups to each other.
• Part-to-whole: "Engines make up 3/5 of the fleet" compares one group to the total.

A part-to-part ratio of 3:2 means the whole has 3 + 2 = 5 parts, so engines are 3/5 of the fleet and trucks are 2/5. Mixing up these two ideas is the most common ratio mistake.

The Ratio-Split Method

Problem: A recruit class of 48 has a ratio of 5 day-shift trainees to 3 night-shift trainees. How many are on the night shift?

1. Add the ratio parts: 5 + 3 = 8 parts.
2. Find one part: 48 ÷ 8 = 6 trainees per part.
3. Multiply by the part asked for: night shift = 3 × 6 = 18 (day shift = 5 × 6 = 30).
4. Check: 30 + 18 = 48, and 30:18 reduces to 5:3.

The trap answer is usually the number from the wrong side of the ratio (30), or a result from dividing by one ratio number instead of the total parts.

Simplifying and Scaling Ratios

• Simplify by dividing both terms by their greatest common factor: 24:36 → divide by 12 → 2:3.
• Scale a ratio by multiplying both terms by the same number. A foam-to-water mix of 3:97 scales to 6:194 or 30:970. The proportion stays the same.

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Proportions: Setting Two Ratios Equal

A proportion says two ratios are equal: a/b = c/d. Solve it by cross-multiplying: a × d = b × c.

Problem: A map scale reads 2 inches = 500 feet. Two hydrants are 7 inches apart on the map. How far apart are they?

2 / 500 = 7 / x
2x = 500 x 7 = 3,500
x = 1,750 feet

Unit-rate shortcut: 500 ÷ 2 = 250 feet per inch, so 7 × 250 = 1,750 feet. Setting up the proportion and finding the unit rate are the same idea. Use whichever feels faster.

Direct vs. Inverse Proportion

• Direct: both quantities rise together. Twice as many gallons weigh twice as much, and more hours at a steady rate cover more miles.
• Inverse: one rises as the other falls. More crews finish a job in less time, and a larger pulley turns more slowly.

Inverse problem: Three crews can clear a debris field in 8 hours. How long would 4 crews take at the same rate?

• Total work = 3 crews × 8 hours = 24 crew-hours.
• 24 ÷ 4 crews = 6 hours.

A candidate who sets this up as a direct proportion gets 10⅔ hours. That is the trap answer, and it fails the common-sense check: more crews should take less time, not more.

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Order of Operations

Every algebra and formula problem depends on doing operations in the right order. The standard sequence is PEMDAS:

1. Parentheses
2. Exponents
3. Multiplication and Division, from left to right
4. Addition and Subtraction, from left to right

Example: 6 + 4 × (5 − 2)² = 6 + 4 × 9 = 6 + 36 = 42. Working left to right without the rules gives 6 + 4 = 10, then 10 × 9 = 90, which is wrong.

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Solving Linear Equations

The goal is to get the variable alone by doing the same operation to both sides, undoing operations in reverse order.

One-step: x − 17 = 42 → add 17 to both sides → x = 59.

Two-step: 3x + 12 = 57

3x + 12 - 12 = 57 - 12   ->  3x = 45
3x / 3 = 45 / 3          ->  x = 15

Variables on both sides: 5x − 8 = 2x + 19

5x - 2x = 19 + 8   ->  3x = 27  ->  x = 9

Always check by substituting: 5(9) − 8 = 37, and 2(9) + 19 = 37. ✔

Translating Words into Equations

Word problem: A hose bed holds 1,000 feet of hose made up of 100-foot and 50-foot lengths. There are twice as many 100-foot lengths as 50-foot lengths. How many 50-foot lengths are there?

• Let x = the number of 50-foot lengths, so there are 2x hundred-foot lengths.
• 50x + 100(2x) = 1,000
• 50x + 200x = 1,000
• 250x = 1,000 → x = 4 fifty-foot lengths (and 8 hundred-foot lengths)
• Check: 4 × 50 + 8 × 100 = 200 + 800 = 1,000 feet. ✔

Fractional counts are a warning sign. If a problem that counts people, lengths, or vehicles gives you 4⅔, your equation is almost certainly set up wrong. Recheck it before choosing an answer.

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Using Formulas: Substitution

Many FCTC-style math items hand you a formula and ask you to plug in values. The skill is careful substitution plus order of operations.

Temperature conversion: F = (9/5 × C) + 32. Convert 40°C:

F = (9/5 x 40) + 32 = 72 + 32 = 104 degrees F

Rearranging a formula: From Area = Length × Width, Width = Area ÷ Length. A 1,440-square-foot room that is 48 feet long is 1,440 ÷ 48 = 30 feet wide.

Averages as Algebra

The mean is the sum divided by the count. A common question asks for the missing value needed to reach an average.

Problem: Your first four practice-test scores are 68, 74, 71, and 77. What must you score on the fifth to average 74?

• Needed total = 74 × 5 = 370.
• Current total = 68 + 74 + 71 + 77 = 290.
• Needed score = 370 − 290 = 80.

Quick-Check Habits

1. Estimate first. Decide whether the answer should be bigger or smaller, and roughly by how much.
2. Substitute your answer back into the original words, not just the equation you wrote.
3. Label units at every step: feet, hours, gallons, crews.
4. Reread what is asked. The question may want the other part of the ratio, or the total rather than the part.