Equivalent representations of rational numbers
Three names for one number
A rational number is any number that can be written as one whole number over another, such as 3/4 or 7/5. Most rational numbers can be written in several ways that look different but mean the same thing. Three quarters of a pizza can be called the fraction 3/4, the decimal 0.75, or the percentage 75%, and not one slice has changed. The number line is the clearest way to see this: each of those three names points to exactly the same place, three quarters of the way from 0 to 1.
Reading the same point three ways
On a line marked from 0 to 1, the halfway point can be written as 1/2, as 0.5, or as 50%, and all three describe the same position. The fraction records how the gap was cut into equal parts. The decimal expresses the same amount using tenths and hundredths. The percentage rescales the whole journey from 0 to 1 into one hundred even steps, so half the journey is fifty of them. Different language, one destination. Once you can hear three names for a single place, moving between them stops being a trick and becomes a translation.
Why equivalent fractions are equal
Equivalent fractions are the part learners most often doubt, because 1/2 and 4/8 look so different on the page. The picture settles it. Shade half of a bar, then slice that same bar into more pieces: cut it into four and two of the four are shaded, cut it into eight and four of the eight are shaded, but the shaded edge never moves. That fixed boundary is why the fractions are equal. Multiplying the top and bottom by the same number simply cuts every piece into smaller pieces and takes proportionally more of them, so 1/2, 2/4 and 4/8 all land on one point.
Converting between the forms
Moving between the forms rests on two simple moves. To turn a fraction into a decimal, divide the top by the bottom: 3 divided by 4 gives 0.75. To turn a decimal into a percentage, multiply by 100, because a percentage counts parts per hundred, so 0.75 becomes 75%. To travel the other way, divide the percentage by 100 to recover the decimal. None of these moves changes the value; they only change the name. Being fluent in all three lets you pick whichever form makes a problem easiest, which is exactly what ratios and rates will ask of you later in the year.
Numbers beyond one
Rational numbers are not trapped between 0 and 1. The value 1.25 means one whole and one quarter, so it sits between 1 and 2, a quarter of the way along. The improper fraction 5/4 names the very same point, because four quarters make one whole and one more quarter carries you past it; the mixed number 1 and 1/4 says the same thing in plainer words. Reading a line in both directions, as whole numbers and as the parts between them, is what lets you place, compare and order any rational number with confidence rather than guesswork.
Where equivalent forms lead
Seeing one rational number behind its many names is groundwork for much of the mathematics that follows. Adding and subtracting fractions depends on rewriting them with a common denominator, an equivalence move. Percentages, discounts and interest all read a fraction as parts per hundred. Ratios and rates compare quantities that are often easiest to handle as decimals or percentages. From here the same number line carries negative rationals, then the coordinates of algebra. The habit built in this unit, treating a fraction, a decimal and a percentage as one value wearing three coats, quietly supports all of it.