How to Solve Proportions: Cross Multiplication, Word Problems, and Solving for x

Map key: 2 cm for 60 km; your trail measures 5 cm. Knowing how to solve proportions means finishing this sentence: 2 is to 60 as 5 is to $x$. The definition: a proportion is an equation $\frac{a}{b}=\frac{c}{d}$, with $b\neq 0$, $d\neq 0$ - Ray's New Higher Algebra (1866): proportion is an equality of ratios.

The X above is the whole method: set up, cross multiply, divide, check.

This page runs them on bare fractions, in word problems, on proportions solve for x stems.

Clearing fractions with the LCD is what classrooms call cross multiplication. The license is Ray's Proposition I (p. 240) - which is why how to solve proportions takes one trade. Multiply $\frac{a}{b}=\frac{c}{d}$ by $bd$:

$\frac{a}{b}\cdot bd=\frac{c}{d}\cdot bd\quad\Longrightarrow\quad ad=bc$

How to solve proportions starts from this trade: fractions out, ordinary equation in.

Numbered, how to solve proportions fits on an index card:

1. Cross multiply. Diagonal products, set equal. From $\frac{x}{6}=\frac{10}{12}$ write $12x = 60$.
2. Divide. $x = \frac{60}{12} = 5$.
3. Check. $\frac{5}{6}=\frac{10}{12}$ - true, so the answer stands. How to solve proportions ends with proof, not hope.

Marisol's $\frac{144}{a}=\frac{9}{4}$ goes the LCD way - $4\cdot 144 = 9a$, same equation, different bookkeeping. These three moves are all how to solve proportions ever uses; the examples below rerun it on fresh numbers.

Problem. Solve the proportion $\frac{y}{96}=\frac{5}{6}$ - the first shape you meet in how to solve proportions.

Step 1, cross multiply. The standard opener in how to solve proportions - each denominator multiplies the opposite numerator: $6y = 96 \times 5 = 480$.

Step 2, divide. $y = \frac{480}{6} = 80$.

Step 3, check. $\frac{80}{96}$ reduces by 16 to $\frac{5}{6}$ - both sides name the same ratio.

One unknown, all else in view - the plainest run.

Problem. Solve the proportion $\frac{120}{a}=\frac{5}{4}$ - the version of how to solve proportions where the letter hides downstairs.

Step 1, cross multiply. $5a = 120 \times 4 = 480$.

Step 2, divide. $a = \frac{480}{5} = 96$.

Step 3, check. $\frac{120}{96}$ reduces by 24 to $\frac{5}{4}$ - true, and $a = 96$ leaves every denominator alive.

A denominator seat changes nothing: all four slots get equal treatment. It only adds a duty: no denominator may hold $0$ - confirm the answer stayed out. Downstairs seats are simply where how to solve proportions asks for one extra look.

How to find proportion begins with a yes-or-no decision: two ratios are in proportion exactly when their cross products are equal. Take $\frac{6}{10}$ and $\frac{21}{35}$: $6\times 35 = 210$ and $10\times 21 = 210$ - equal products, a proportion. Now $\frac{2}{3}$ and $\frac{9}{14}$: $2\times 14 = 28$ but $3\times 9 = 27$ - no proportion. One trap: $(3, 4)$ and $(15, 16)$ both differ by 1, yet $\frac{3}{4}\neq\frac{15}{16}$ - subtraction settles nothing; cross products do.

The setup half is the doorway into how to solve proportions from sentences: two fractions, units aligned - calories over ounces, height over shadow. Two terms come from the sentence, one from the question; the units decide who sits over whom. From here on, how to solve proportions from word problems is bookkeeping. Most failures in how to find proportion happen at setup: label first, multiply second - after that, how to solve proportions is pure arithmetic.

Problem. An 8-ounce serving of a sports drink has 60 calories. How many calories are in a 20-ounce bottle?

Step 1, name the unknown. Let $c$ = calories in the bottle - naming first, as always in how to solve proportions from sentences.

Step 2, set up with units aligned - calories over ounces on both sides; alignment is where how to solve proportions from sentences is won or lost:

$\frac{60}{8}=\frac{c}{20}$

Step 3, cross multiply. $8c = 60 \times 20 = 1200$.

Step 4, divide and answer. $c = 150$ - the bottle holds 150 calories.

Four moves take a sentence to a proportion; from there, how to solve proportions finishes as cleanly as on bare numbers. Numbers cleaned from a drink-calorie word problem.

When the missing term wears the letter, proportions solve for x in one of two shapes - and how to solve proportions runs the same habit on both: cross, divide, check. Two shapes, one crossing:

One guard: neither $b$ nor $d$ may be $0$. Apart from that, this corner of how to solve proportions is a two-step job.

Two mini runs. Run A. $\frac{x}{45}=\frac{2}{5}$: cross to $5x=90$, so $x=18$, and $\frac{18}{45}=\frac{2}{5}$ holds. Run B. $\frac{15}{x}=\frac{3}{4}$: cross to $3x=60$, so $x=20$, and $\frac{15}{20}=\frac{3}{4}$ confirms it - proportions solve for x from a denominator seat just as smoothly. Both runs finish in three lines; with a letter, how to solve proportions is exactly this short.

Per. Miles per gallon - a rate with per is one ratio waiting for a second. For every. 2 cm for every 60 km. Same moment. Shadows cast at the same time - two situations locked to one rate.

Spot the signal and name the unknown; how to solve proportions then starts exactly as in the examples above: units aligned, cross, divide, check. Each signal hands how to solve proportions its raw material - two known terms and a question. Learn the three signals and how to solve proportions never gets ambushed by a sentence.

Cross products never ask for whole numbers. From $\frac{2.7}{j}=\frac{0.9}{0.2}$, multiply across: $0.9j = 0.54$, so $j = 0.6$. Check: $\frac{2.7}{0.6} = 4.5$ and $\frac{0.9}{0.2} = 4.5$ - equal ratios, confirmed. Dot or no dot, how to solve proportions runs identically. The decimal point just sits there while the loop runs - multiply, divide, check. Dot placement is notation, not method. The run above closed on one division with the dots still in place, and no decimal seat ever changes how to solve proportions. Dots are luggage, not obstacles, for how to solve proportions.

A slipped crossing still produces a plausible number - hence the third move. Substitution is where how to solve proportions catches its own mistakes: put the answer back and the equal-ratio claim holds or collapses. Run it every time - it is the moment how to solve proportions proves its answer; Plausible-but-wrong is the failure mode the check exists to catch. This five-second habit is what makes how to solve proportions safe to use.

Problem. Solve the proportion $\frac{x}{56}=\frac{4}{7}$ - the standard shape proportions solve for x in, and the fastest run in how to solve proportions.

Step 1, cross multiply. $7x=56\times4=224$.

Step 2, divide. $x=\frac{224}{7}=32$.

Step 3, check. $\frac{32}{56}$ reduces by 8 to $\frac{4}{7}$ - both sides name the same ratio, the check that certifies how to solve proportions.

Two steps, one check, done - upstairs or downstairs. Substituting back closes the case; that uniformity is why how to solve proportions is a one-method subject.

How to solve proportions in one line: two ratios, one equals sign, units aligned - cross multiply the diagonals, divide, check by substitution. Run it on bare fractions, in word problems, in every proportions solve for x stem. From maps to recipes to decimals, how to solve proportions is the same three moves in every costume. That loop is the whole method.

1. Adding across instead of multiplying. From $\frac{2}{3}=\frac{x}{9}$, writing $2+3 = x+9$ invents a rule nobody proved. The first rule of how to solve proportions: multiply, never add. So $3x = 18$, $x = 6$, and the check $\frac{6}{9}=\frac{2}{3}$ confirms it.

2. Pairing the wrong diagonals. In $\frac{a}{b}=\frac{c}{d}$ each term pairs across the equals sign: $ad = bc$. Writing $ab = cd$ pairs neighbors, not diagonals - trace the X first: how to solve proportions starts with the right diagonals.

3. Mixing up the unit slots. From Example 3, $\frac{60}{8}=\frac{c}{20}$ works because calories sit over ounces on both sides. Flip one side and the answer comes out inverted - the classic way how to find proportion goes wrong.

4. Forgetting to distribute. From $6(k+15) = 4(2k+3)$, the slip writes $6k + 15$. Both terms take the multiplier: $6k + 90 = 8k + 12$. Bracketed how to solve proportions runs die here most often.

5. Distrusting fractional answers. Solve $\frac{7}{c}=\frac{4}{9}$: cross multiplication gives $4c = 63$, so $c = \frac{63}{4}$. Fractional endings are normal - a complete answer is a complete answer.