How to Find Proportional Ratios

How to Find Proportional Ratios

 Read,3 minutes

What Are Proportional Ratios?

Two ratios are proportional when they name the same comparison. For example, \(2:3\) and \(8:12\) are proportional because both simplify to \(2:3\).

Using Cross Products

For a proportion \(\frac{a}{b}=\frac{c}{d}\), the cross products must be equal: \(ad=bc\).

Solving a Proportion

To solve for an unknown, cross multiply, simplify the equation, and divide to isolate the variable. Example: \(\frac{x}{15}=\frac{4}{5}\). Cross multiply to get \(5x=60\), so \(x=12\).

Proportional Ratios

Think of this lesson as more than a rule to memorize. Proportional Ratios 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 Proportional Ratios

1) Are the ratios \(3:5\) and \(6:10\) proportional?

2) Are the ratios \(4:9\) and \(12:28\) proportional?

3) Solve \(\frac{x}{6}=\frac{5}{3}\).

4) Solve \(\frac{8}{x}=\frac{2}{5}\).

5) Solve \(7:12=x:36\).

6) Solve \(15:20=9:x\).

7) Find \(x\) if \(\frac{x+1}{4}=\frac{3}{2}\).

8) Find \(x\) if \(\frac{10}{x-2}=\frac{5}{6}\).

9) A model car is built at a scale of \(1:24\). If the model is \(7\) inches long, how long is the real car in inches?

10) A recipe uses \(3\) cups of oats for \(8\) servings. How many cups are needed for \(20\) servings?

11) The ratios \(2x:15\) and \(6:9\) are proportional. Find \(x\).

12) The ratios \(x+3:18\) and \(5:6\) are proportional. Find \(x\).

13) Find \(y\) if the ordered pairs \((4,10)\) and \((14,y)\) are in the same proportional relationship.

14) A graph of a proportional relationship passes through \((6,15)\). What is the constant of proportionality \(k\) in \(y=kx\)?

15) Use the constant from \((6,15)\) to find \(y\) when \(x=22\).

16) Solve \(\frac{3x-2}{10}=\frac{7}{5}\).

17) Solve \(\frac{2x+5}{9}=\frac{x+7}{6}\).

18) A shadow problem has proportional heights and shadow lengths. A \(5\)-ft person casts an \(8\)-ft shadow. A tree casts a \(30\)-ft shadow. How tall is the tree?

19) A mixture keeps acid to water in the ratio \(2:7\). How many liters of water are needed with \(11\) liters of acid?

20) Two ratios are proportional: \(\frac{x-4}{x+2}=\frac{3}{5}\). Find \(x\).

 
1) Compare by simplifying or cross multiplying. \(3\cdot10=30\) and \(5\cdot6=30\). Since the cross products match, the ratios are proportional.
2) Cross multiply: \(4\cdot28=112\) and \(9\cdot12=108\). The products are not equal, so the ratios are not proportional.
3) Cross multiply: \(3x=6\cdot5=30\). Divide by \(3\): \(x=10\).
4) Cross multiply: \(8\cdot5=2x\), so \(40=2x\). Divide by \(2\): \(x=20\).
5) Write \(\frac{7}{12}=\frac{x}{36}\). Cross multiply: \(12x=7\cdot36=252\). Then \(x=21\).
6) Write \(\frac{15}{20}=\frac{9}{x}\). Cross multiply: \(15x=20\cdot9=180\). Then \(x=12\).
7) Cross multiply: \(2(x+1)=4\cdot3\). So \(2x+2=12\), \(2x=10\), and \(x=5\).
8) Cross multiply: \(10\cdot6=5(x-2)\). So \(60=5x-10\), \(70=5x\), and \(x=14\).
9) The scale \(1:24\) means real length is \(24\) times the model length. \(7\cdot24=168\). The real car is \(168\) inches long.
10) Set \(\frac{3}{8}=\frac{x}{20}\). Cross multiply: \(8x=60\). Therefore \(x=7.5\). The recipe needs \(7.5\) cups.
11) Set \(\frac{2x}{15}=\frac{6}{9}\). Cross multiply: \(9(2x)=15\cdot6\), so \(18x=90\). Thus \(x=5\).
12) Set \(\frac{x+3}{18}=\frac{5}{6}\). Cross multiply: \(6(x+3)=90\). Then \(6x+18=90\), \(6x=72\), and \(x=12\).
13) In a proportional relationship, \(\frac{y}{x}\) is constant. From \((4,10)\), \(k=\frac{10}{4}=2.5\). For \(x=14\), \(y=2.5\cdot14=35\).
14) Use \(y=kx\). Substitute \((6,15)\): \(15=6k\). Thus \(k=\frac{15}{6}=\frac{5}{2}\).
15) From the previous constant, \(k=\frac{5}{2}\). Then \(y=\frac{5}{2}\cdot22=55\).
16) Cross multiply: \(5(3x-2)=10\cdot7\). So \(15x-10=70\), \(15x=80\), and \(x=\frac{16}{3}\).
17) Cross multiply: \(6(2x+5)=9(x+7)\). Then \(12x+30=9x+63\), so \(3x=33\), and \(x=11\).
18) Set height over shadow equal: \(\frac{5}{8}=\frac{x}{30}\). Cross multiply: \(8x=150\). Therefore \(x=18.75\). The tree is \(18.75\) ft tall.
19) Use acid to water \(2:7\). Set \(\frac{2}{7}=\frac{11}{w}\). Cross multiply: \(2w=77\), so \(w=38.5\). Use \(38.5\) liters of water.
20) Cross multiply: \(5(x-4)=3(x+2)\). Then \(5x-20=3x+6\), so \(2x=26\), and \(x=13\).

Proportional Ratios Quiz