You're on a ruck march. Two hours in, you've covered 6 miles. The objective is still 15 miles away. If you hold this pace, how much longer until you get there?
Most people guess. "Three hours? Maybe four?" The problem with a guess is that you plan water, daylight, and morale around a number you made up. Ratios, rates, and proportions give you a way to replace the guess with an answer you can defend. The same skills show up on a job site when you mix concrete, at the gas pump when you work out fuel, and in the gym when you track reps per minute.
By the end of this guide you'll be able to write a ratio three different ways and simplify it, calculate a unit rate from two quantities with different units, set up a proportion with matching units in matching positions, and solve it using cross multiplication. And you'll know the answer to the ruck march: 5 hours.
No background needed. We build everything from scratch.
ð What Is a Ratio?
A ratio is a comparison of two quantities using division. That's the whole definition.
You can write the same ratio three different ways:
\[3:2 \qquad \frac{3}{2} \qquad 3 \text{ to } 2\]All three say the same thing. Which form you use depends on the situation, but you should recognize all of them.
The one rule you cannot skip is that order matters. Say a platoon has 36 infantry personnel and 24 support personnel. The first number always belongs to infantry and the second always belongs to support. So "infantry to support" is \(36:24\), and "support to infantry" is \(24:36\). Those are different statements. Flip the order and you've changed the meaning, and any math you build on top of it will be wrong.
✂️ Simplifying Ratios with the Greatest Common Factor
A ratio like \(36:24\) is hard to picture. Simplified, it's easy to read. To simplify, divide both numbers by the same number, specifically the greatest common factor (GCF), the largest number that divides evenly into both.
For 36 and 24, the GCF is 12:
\[\frac{36 \div 12}{24 \div 12} = \frac{3}{2}\]So the ratio of infantry to support is \(3:2\). For every 3 infantry, there are 2 support personnel. That's much easier to take in than 36 and 24.
Notice that we're allowed to simplify here because both numbers share the same unit: personnel. Infantry and support are two types of personnel, but they're both counted as people. That matters, because it's the difference between a ratio and a rate, which we'll get to next.
⛽ Rates and Unit Rates
A rate is a ratio that compares quantities with different units. Think miles per gallon, dollars per pound, or reps per minute.
The word per is your guide. It separates the top of the fraction from the bottom. Whatever comes before "per" goes on top, and whatever comes after goes on the bottom.
- Miles per gallon: \(\dfrac{\text{miles}}{\text{gallon}}\)
- Dollars per pound: \(\dfrac{\text{dollars}}{\text{pound}}\)
- Reps per minute: \(\dfrac{\text{reps}}{\text{minute}}\)
A unit rate is a rate with a 1 in the denominator. It tells you how much of one thing you get per one of the other. "60 miles per hour" is really:
\[\frac{60 \text{ miles}}{1 \text{ hour}}\]Same with chicken at $5 per pound, which is \(\dfrac{\$5}{1 \text{ lb}}\). (Yes, that's expensive.)
Unit rates are useful because they put different quantities on the same footing. Two packages of ground beef at different weights and prices are hard to compare directly. Divide each price by its weight and you have dollars per pound for both, and the better deal is obvious. The same move works for comparing two trucks on fuel efficiency or two workouts on reps per minute: reduce each to "per one," then compare.
Worked example: fuel efficiency
A truck drives 312 miles on 12 gallons of fuel. What's its fuel efficiency, in miles per gallon?
"Miles" comes before "per," so it goes on top. "Gallons" goes on the bottom:
\[\frac{312 \text{ miles}}{12 \text{ gallons}} = 26 \text{ miles per gallon}\]Always attach the units to your final answer. "26" alone doesn't tell the reader anything. "26 miles per gallon" does.
⚖️ What Is a Proportion?
A proportion is two ratios set equal to each other. It says the relationship on the left side is the same as the relationship on the right.
There's one condition that makes proportions work: the same quantity must sit in the same position on both sides. If miles are on top on the left, miles must be on top on the right. If hours are on the bottom on the left, hours must be on the bottom on the right.
Back to the ruck march. We know 6 miles took 2 hours, and we want to know how many hours (call that \(x\)) it will take to cover 15 miles:
\[\frac{6 \text{ miles}}{2 \text{ hours}} = \frac{15 \text{ miles}}{x \text{ hours}}\]Miles on top, hours on the bottom, both sides. That's a valid setup.
There's a fast way to see the answer. \(6 \div 2 = 3\), so you're moving at 3 miles per hour. For the right-hand side to also equal 3, you need \(15 \div x = 3\), which means \(x = 5\). Five hours.
That shortcut works when the numbers are friendly. When they aren't, you need a method that works every time, and that's cross multiplication.
✖️ Cross Multiplication: The Reliable Way to Solve Proportions
Here's the rule. If
\[\frac{a}{b} = \frac{c}{d}\]then
\[a \cdot d = b \cdot c\]Picture an X across the equals sign. You multiply along each diagonal, and the two products are equal.
A quick example
Solve \(\dfrac{4}{9} = \dfrac{x}{45}\).
Cross multiply:
\[9x = 4 \cdot 45 = 180\]Divide both sides by 9:
\[x = 20\]Check it. To get from 9 to 45, you multiply by 5. So you should be able to multiply the 4 by 5 too: \(4 \cdot 5 = 20\). ✅
Checking your work like this takes ten seconds and catches almost every setup error.
Back to the ruck march
Now we can solve the opening problem the reliable way:
\[\frac{6}{2} = \frac{15}{x}\] \[6x = 2 \cdot 15 = 30\] \[x = 5\]It will take 5 more hours to reach the objective. No guessing.
ð§ą Worked Example: Concrete Mix
Construction math is mostly proportions. Say a mix calls for 2 bags of cement for every 7 bags of sand. If you're using 10 bags of cement, how many bags of sand do you need?
Put cement on top and sand on the bottom, on both sides:
\[\frac{2 \text{ cement}}{7 \text{ sand}} = \frac{10 \text{ cement}}{x \text{ sand}}\]Cross multiply:
\[2x = 7 \cdot 10 = 70\]Divide both sides by 2:
\[x = 35\]You need 35 bags of sand.
Sanity check: you went from 2 bags of cement to 10, which is 5 times as much. So the sand should also be 5 times as much: \(7 \cdot 5 = 35\). ✅
ðŠĪ The Trap Problem: What Happens When You Flip One Side
Here's where people lose points, and where people lose real-world accuracy. Go back to the ruck march and set it up wrong, with hours on top on one side and miles on top on the other:
\[\frac{6 \text{ miles}}{2 \text{ hours}} = \frac{x \text{ hours}}{15 \text{ miles}}\]The algebra works perfectly. Cross multiply:
\[6 \cdot 15 = 2 \cdot x\] \[90 = 2x\] \[x = 45\]So the math says it takes 45 hours to march 15 miles. But you just covered 6 miles in 2 hours. Moving 15 miles in 45 hours would mean you're slower than a toddler. That doesn't pass the smell test.
The algebra didn't fail. The setup did. Cross multiplication gives you a correct answer to the question you set up, even when you set up the wrong question.
Make it a habit, every time. Before you solve, ask whether miles are on top on both sides and hours on the bottom on both sides. After you solve, ask whether the answer makes sense. A rational decision beats a mechanical one.
⚠️ Five Common Mistakes with Ratios and Proportions
1. Writing the ratio in the wrong order. Infantry to support (\(3:2\)) is not support to infantry (\(2:3\)). Decide which quantity comes first and keep that order on both sides of a proportion.
2. Cross multiplying straight across. Cross multiplication means an X, not a multiplication of the two tops and the two bottoms. For \(\dfrac{3}{5} = \dfrac{x}{20}\), the correct move is:
\[3 \cdot 20 = 5x \quad\Rightarrow\quad 60 = 5x \quad\Rightarrow\quad x = 12\]3. Mismatched units. Say you want the ratio of 2 feet to 18 inches. It is not \(2:18\), because those are different units. Convert first. There are 12 inches in a foot, so 2 feet is \(2 \cdot 12 = 24\) inches. Now both quantities are in inches:
\[24 : 18 = \frac{24 \div 6}{18 \div 6} = \frac{4}{3}\]The ratio is \(4:3\).
4. Forgetting parentheses. When a proportion has parentheses, you have to distribute. For example, \(6(x + 3)\) becomes:
\[6(x + 3) = 6x + 18\]and not \(6x + 3\). The 6 multiplies everything inside.
5. Skipping the smell test. The 45-hour march is the perfect example. If an answer looks absurd, go back and check your setup.
✅ A Four-Question Setup Checklist
Every proportion problem, from the ruck march to the concrete mix, can be run through the same four questions before you touch a calculator:
- What are the two quantities? (Miles and hours. Cement and sand.)
- Which one is on top? Pick it once and keep it there on both sides.
- Which value is unknown? Give it a variable, usually \(x\), and put it in the matching position.
- Does the answer pass the smell test? If more miles should take more time, your answer should show that.
Thirty seconds on that checklist prevents the 45-hour march.
ð️ How to Practice
Reading about proportions only gets you part of the way. Work these five, getting harder as you go. Label your units, cross multiply, and run the smell test.
- Simplify the ratio \(20:35\).
- A runner covers 6 miles in 48 minutes. How many minutes per mile?
- Solve: \(\dfrac{3}{5} = \dfrac{x}{20}\)
- A truck uses 5 gallons of fuel to drive 130 miles. At the same rate, how many gallons does it need for 390 miles?
- Challenge: Solve \(\dfrac{x + 3}{4} = \dfrac{9}{6}\)
Answer key:
- \(20:35 \to \dfrac{20 \div 5}{35 \div 5} = 4:7\)
- \(48 \div 6 = 8\) minutes per mile
- \(3 \cdot 20 = 5x \Rightarrow 60 = 5x \Rightarrow x = 12\). Check: \(\tfrac{3}{5} = 0.6\) and \(\tfrac{12}{20} = 0.6\) ✅
- \(\dfrac{5}{130} = \dfrac{x}{390} \Rightarrow 130x = 1950 \Rightarrow x = 15\) gallons. Check: 390 is 3 times 130, and 15 is 3 times 5 ✅
- \(6(x + 3) = 4 \cdot 9 \Rightarrow 6x + 18 = 36 \Rightarrow 6x = 18 \Rightarrow x = 3\). Check: \(\tfrac{3 + 3}{4} = 1.5\) and \(\tfrac{9}{6} = 1.5\) ✅
ðĨ The Takeaway
Everything in this guide comes down to four moves. A ratio compares two quantities, in a specific order. A rate compares different units, and a unit rate puts a 1 in the denominator. A proportion sets two ratios equal, with the same quantity in the same position on both sides. And cross multiplication solves for the unknown, as long as you set it up right and sanity-check the answer.
Master those and you'll handle ratios, rates, and proportions problems from the road march to the job site to the gym, without guessing. Work the practice problems above, and keep the smell test in your back pocket.
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