How to Create a Proportion
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A proportion is an equation that says two ratios are equal. For example, \(\frac{2}{5}=\frac{8}{20}\) is a proportion because both ratios simplify to the same value.
To create a proportion, keep the same type of quantity in the same position in both ratios. If the first ratio is dollars over tickets, the second ratio should also be dollars over tickets.
You can complete a proportion by scaling both parts of a ratio by the same factor or by using cross products. In \(\frac{a}{b}=\frac{c}{d}\), the cross products are equal: \(ad=bc\).
Create a Proportion
Think of this lesson as more than a rule to memorize. Create a Proportion is about comparisons, scaling, and equal ratios. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
A ratio compares quantities, and a proportion says two ratios are equal. Cross products help because \(\frac{a}{b}=\frac{c}{d}\) implies \(ad=bc\).
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Read what is given and what is being asked.
- Choose the rule that connects them.
- Substitute carefully and simplify in small steps.
- Check the final answer against the original question.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for Create a Proportion
1) Create a proportion equivalent to \(\frac{2}{5}\) using denominator \(20\).
2) Create a proportion equivalent to \(\frac{3}{7}\) using numerator \(15\).
3) Complete the proportion: \(\frac{4}{9}=\frac{x}{27}\).
4) Complete the proportion: \(\frac{6}{11}=\frac{30}{x}\).
5) Create a proportion for \(\$18\) for \(3\) tickets and the cost \(x\) for \(8\) tickets.
6) Create a proportion for \(120\) miles in \(2\) hours and \(m\) miles in \(5\) hours.
7) Complete the proportion: \(\frac{5}{8}=\frac{x}{40}\).
8) Complete the proportion: \(\frac{9}{14}=\frac{45}{x}\).
9) Use cross products to solve: \(\frac{7}{12}=\frac{x}{36}\).
10) Use cross products to solve: \(\frac{10}{15}=\frac{18}{x}\).
11) Create and solve a proportion: \(4\) pounds of apples cost \(\$7.20\). How much do \(10\) pounds cost?
12) Create and solve a proportion: \(15\) laps take \(6\) minutes. How long do \(40\) laps take at the same pace?
13) Complete the proportion: \(\frac{x}{18}=\frac{14}{21}\).
14) Complete the proportion: \(\frac{16}{x}=\frac{28}{35}\).
15) Create a proportion to convert \(45\) feet to yards, using \(3\) feet per \(1\) yard, then solve.
16) A scale map uses \(2\) centimeters for \(15\) kilometers. Create and solve a proportion for \(9\) centimeters.
17) Solve: \(\frac{x+2}{6}=\frac{5}{3}\).
18) Solve: \(\frac{2x-1}{9}=\frac{7}{3}\).
19) A model car is built at a scale of \(1:24\). Create and solve a proportion for a model length of \(7.5\) inches.
20) A mixture uses concentrate and water in a ratio of \(3:8\). Create and solve a proportion for the amount of water needed with \(10.5\) cups of concentrate.
1) Set \(\frac{2}{5}=\frac{x}{20}\). Since \(5\cdot 4=20\), multiply the numerator by \(4\) too: \(2\cdot 4=8\). A correct proportion is \(\color{red}{\frac{2}{5}=\frac{8}{20}}\).
2) Set \(\frac{3}{7}=\frac{15}{x}\). Since \(3\cdot 5=15\), multiply the denominator by \(5\): \(7\cdot 5=35\). A correct proportion is \(\color{red}{\frac{3}{7}=\frac{15}{35}}\).
3) The denominator changes from \(9\) to \(27\), which is multiplying by \(3\). Multiply the numerator by \(3\): \(4\cdot 3=12\). So \(x=\color{red}{12}\).
4) The numerator changes from \(6\) to \(30\), which is multiplying by \(5\). Multiply the denominator by \(5\): \(11\cdot 5=55\). So \(x=\color{red}{55}\).
5) Keep dollars over tickets in both ratios: \(\frac{18}{3}=\frac{x}{8}\). This compares the same quantities in the same order. The proportion is \(\color{red}{\frac{18}{3}=\frac{x}{8}}\).
6) Use miles over hours for both ratios: \(\frac{120}{2}=\frac{m}{5}\). The units match across the numerators and denominators, so the proportion is \(\color{red}{\frac{120}{2}=\frac{m}{5}}\).
7) The denominator changes from \(8\) to \(40\), a factor of \(5\). Multiply the numerator by \(5\): \(5\cdot 5=25\). So \(x=\color{red}{25}\).
8) The numerator changes from \(9\) to \(45\), a factor of \(5\). Multiply the denominator by \(5\): \(14\cdot 5=70\). So \(x=\color{red}{70}\).
9) Cross multiply: \(7\cdot 36 = 12x\). Then \(252=12x\). Divide by \(12\): \(x=21\). So \(x=\color{red}{21}\).
10) Cross multiply: \(10x=15\cdot 18\). Then \(10x=270\). Divide by \(10\): \(x=27\). So \(x=\color{red}{27}\).
11) Use cost over pounds: \(\frac{7.20}{4}=\frac{x}{10}\). Cross multiply: \(4x=72\). Divide by \(4\): \(x=18\). The cost is \(\color{red}{\$18.00}\).
12) Use minutes over laps: \(\frac{6}{15}=\frac{x}{40}\). Cross multiply: \(15x=240\). Divide by \(15\): \(x=16\). It takes \(\color{red}{16}\) minutes.
13) Simplify \(\frac{14}{21}\) to \(\frac{2}{3}\). Then \(\frac{x}{18}=\frac{2}{3}\). Since \(3\cdot 6=18\), \(2\cdot 6=12\). So \(x=\color{red}{12}\).
14) Simplify \(\frac{28}{35}\) to \(\frac{4}{5}\). Then \(\frac{16}{x}=\frac{4}{5}\). Since \(4\cdot 4=16\), multiply \(5\) by \(4\): \(x=20\). So \(x=\color{red}{20}\).
15) Use feet over yards: \(\frac{3}{1}=\frac{45}{y}\). Cross multiply: \(3y=45\). Divide by \(3\): \(y=15\). The length is \(\color{red}{15}\) yards.
16) Use kilometers over centimeters: \(\frac{15}{2}=\frac{x}{9}\). Cross multiply: \(2x=135\). Divide by \(2\): \(x=67.5\). The actual distance is \(\color{red}{67.5}\) kilometers.
17) Cross multiply: \(3(x+2)=6\cdot 5\). Then \(3x+6=30\). Subtract \(6\): \(3x=24\). Divide by \(3\): \(x=\color{red}{8}\).
18) Cross multiply: \(3(2x-1)=9\cdot 7\). Then \(6x-3=63\). Add \(3\): \(6x=66\). Divide by \(6\): \(x=\color{red}{11}\).
19) Use model over actual: \(\frac{1}{24}=\frac{7.5}{x}\). Cross multiply: \(x=24\cdot 7.5=180\). The actual car length is \(\color{red}{180}\) inches, or \(15\) feet.
20) Use concentrate over water: \(\frac{3}{8}=\frac{10.5}{w}\). Cross multiply: \(3w=8\cdot 10.5=84\). Divide by \(3\): \(w=28\). The amount of water is \(\color{red}{28}\) cups.
Create a Proportion Quiz
