312=19x{\displaystyle {\frac {\color {blue}{3}}{\color {blue}{12}}}={\frac {19}{x}}} How are 3 and 12 related? 3×4=12{\displaystyle 3\times {\textbf {4}}=12} The other vertical column is related the same way: 19×4=x{\displaystyle 19\times {\textbf {4}}=x} x=76{\displaystyle x=76}, so 312=1976{\displaystyle {\frac {3}{12}}={\frac {19}{76}}}
x5=3610{\displaystyle {\frac {x}{\color {orange}{5}}}={\frac {36}{\color {orange}{10}}}} What is the relationship between 5 and 10? 5×2=10{\displaystyle 5\times {\textbf {2}}=10} The other horizontal row is related the same way: x×2=36{\displaystyle x\times {\textbf {2}}=36} x=18{\displaystyle x=18}, so 185=3610{\displaystyle {\frac {18}{5}}={\frac {36}{10}}}
14x=46{\displaystyle {\frac {\color {purple}{14}}{\color {green}{x}}}={\frac {\color {green}{4}}{\color {purple}{6}}}}
14×6=84{\displaystyle \color {purple}{14\times 6}\color {black}{=}84}
84÷4=21{\displaystyle \color {purple}{84}\color {black}{\div }\color {green}{4}\color {black}{=21}} x=21{\displaystyle x=21}, so you can fill in your proportion like this: 1421=46{\displaystyle {\frac {14}{21}}={\frac {4}{6}}}
48 128 x 8 Each column in this table represents a fraction. All of the fractions in this table are equal to each other.
For example, try dividing the top and bottom of 1288{\displaystyle {\frac {128}{8}}} by 2. This gives you a new fraction, 644{\displaystyle {\frac {64}{4}}}, to put in your table. 48 64 128 x 4 8
Both the top and bottom of 644{\displaystyle {\frac {64}{4}}} are divisible by 2 again, giving you the fraction 322{\displaystyle {\frac {32}{2}}}. 32 48 64 128 2 x 4 8 The x in your table is somewhere between 2 and 4. Let’s try 3 by plugging it back into your proportion: 483=1288{\displaystyle {\frac {48}{\bf {3}}}={\frac {128}{8}}}
To check whether 483=1288{\displaystyle {\frac {48}{3}}={\frac {128}{8}}} is the correct solution, draw two diagonal lines across the fraction. Multiply the two numbers along one line: 48×8=384{\displaystyle 48\times 8=384}. Now multiply the two numbers along the other line: 3×128=384{\displaystyle 3\times 128=384}. The two answers are the same, which means your answer is correct.
The proportion will always follow the form PartWhole=%100{\displaystyle {\frac {Part}{Whole}}={\frac {%}{100}}}. For word problems, the “part” usually appears next to the word “is”, and the “whole” usually comes after the word “of”. [3] X Research source For example, “3 is what percentage of 6?” can be rewritten as 36=x100{\displaystyle {\frac {3}{6}}={\frac {x}{100}}}. The percentage is unknown, so we write it as x{\displaystyle x} and solve for it.
First multiply across the diagonal line with two known numbers. For the proportion 36=x100{\displaystyle {\frac {\color {purple}{3}}{6}}={\frac {x}{\color {purple}{100}}}}, that means multiplying 3×100=300{\displaystyle 3\times 100=300}. Now divide your answer by the last remaining number in the proportion: 300÷6=50{\displaystyle 300\div 6=50}. x=50{\displaystyle x=50} and the complete proportion is 36=50100{\displaystyle {\frac {3}{6}}={\frac {50}{100}}}
You can change the left hand side of the equation, as long as you do the same math to the right hand side.
For example, start with the proportion 1727=13x{\displaystyle {\frac {17}{27}}={\frac {13}{x}}}. To get rid of the fraction on the left, multiply both sides by 27: 27×1727=27×13x{\displaystyle {\frac {27\times 17}{27}}={\frac {27\times 13}{x}}} The 27s on the left cancel out: 17=27×13x{\displaystyle 17={\frac {27\times 13}{x}}}
x×17=x×27×13x{\displaystyle x\times 17={\frac {x\times 27\times 13}{x}}} The two x{\displaystyle x}s on the right cancel out: 17x=27×13{\displaystyle 17x=27\times 13}
17x17=27×1317{\displaystyle {\frac {17x}{17}}={\frac {27\times 13}{17}}} The 17s on the left cancel out: x=27×1317{\displaystyle x={\frac {27\times 13}{17}}}
x=27×1317=35117{\displaystyle x={\frac {27\times 13}{17}}={\frac {351}{17}}} One big advantage to this method is that it works even when x{\displaystyle x} is a difficult number like this. But if this didn’t make much sense to you, that’s okay: most teachers and textbooks start with the other methods above and teach you algebra a little later.
3x4=48x{\displaystyle {\frac {3x}{4}}={\frac {48}{x}}} Multiply by x{\displaystyle x} on both sides: x×3x4=x×48x{\displaystyle x\times {\frac {3x}{4}}=x\times {\frac {48}{x}}} Simplify: 3x24=48{\displaystyle {\frac {3x^{2}}{4}}=48} Multiply by 4 on both sides: 4×3x24=4×48{\displaystyle 4\times {\frac {3x^{2}}{4}}=4\times 48} Simplify: 3x2=192{\displaystyle 3x^{2}=192} Divide by 3 on both sides: 3x23=1923{\displaystyle {\frac {3x^{2}}{3}}={\frac {192}{3}}} Simplify: x2=64{\displaystyle x^{2}=64} Find the square root: x=64=±8{\displaystyle x={\sqrt {64}}=\pm 8}
Warning: This is a difficult example. If you haven’t learned about quadratic equations yet, you might want to skip this part. 3x+1=2x8{\displaystyle {\frac {3}{x+1}}={\frac {2x}{8}}} Multiply by (x+1){\displaystyle (x+1)}: (x+1)3x+1=2x(x+1)8{\displaystyle (x+1){\frac {3}{x+1}}={\frac {2x(x+1)}{8}}} Simplify. Remember to multiply 2x{\displaystyle 2x} with both terms in the parentheses, and add the results together: 3=2x2+2x8{\displaystyle 3={\frac {2x^{2}+2x}{8}}} The fraction on the right has terms that are all divisible by 2. Simplify: 3=x2+x4{\displaystyle 3={\frac {x^{2}+x}{4}}} Multiply by 4 on both sides: 4×3=4×x2+x4{\displaystyle 4\times 3=4\times {\frac {x^{2}+x}{4}}} Simplify: 12=x2+x{\displaystyle 12=x^{2}+x} Subtract 12 to get zero on one side: x2+x−12=0{\displaystyle x^{2}+x-12=0} You can now solve this as a quadratic equation, using any method that you’ve learned. For example, you can factor this as (x+4)(x−3)=0{\displaystyle (x+4)(x-3)=0}, then solve for x+4=0{\displaystyle x+4=0} and x−3=0{\displaystyle x-3=0} to get your two answers, x=−4{\displaystyle x=-4} and x=3{\displaystyle x=3}.
