Drag 2/7, 0.28 and 28% onto a Number Line — Which One Wins?

You are handed three numbers — \tfrac{2}{7}, 0.28 and 28\% — and asked which is the biggest. They all look like they should be roughly a quarter. Without a calculator, which one wins? This is the kind of question a live number line answers in two seconds: drag each number onto the line, look at the dots, and read off the order from left to right.

The picture

Below is a single number line from 0 to 1. Three draggable dots start stacked on the left. Slide them along the line to the positions you think each of \tfrac{2}{7}, 0.28 and 28\% lives at. The readouts update as you drag, so you can home in on the exact position of each number.

Three draggable dots and a number line from $0$ to $1$. Slide each dot to where you think the corresponding number lives, and watch the readout above. The target value for each dot is listed in the panel — use it to check your placement. $\tfrac{2}{7}$ lands a whisker to the right of $0.28$, and $0.28$ and $28\%$ are the exact same point.

Why the picture works: a number line is a visual ruler for comparing sizes. A number has one place on it. If you put two different-looking expressions — a fraction, a decimal, a percentage — at the same point, it is because they name the same number. If they land at different points, the one further to the right is the larger number, by the definition of the line. No arithmetic is required once the dots are placed.

Reading off the answer

With the dots in place, the order is immediate.

28\% is exactly 0.28. A percentage is just a fraction with denominator 100: 28\% = \tfrac{28}{100} = 0.28. So B and C sit on the same point on the line — they are the same number written two ways.
\tfrac{2}{7} is 0.285714\dots (the repeating block is \overline{285714}). That is a tiny bit bigger than 0.28.

So the order from smallest to biggest is

0.28 = 28\% \;<\; \frac{2}{7}

and the winner is \tfrac{2}{7}, by a margin of about 0.006 — roughly six thousandths of the way along the line. On a line 460 pixels wide, that's under three pixels. Tiny, but visible.

Why this matters more than it looks

A huge part of school arithmetic is silently moving between these three notations. A shopkeeper's "25\% off" sign is a fraction \tfrac{25}{100} is a decimal 0.25 is a point on the line at one quarter of the distance from 0 to 1. They are four faces of the same object, and you will read price tags, probability questions, batting averages and exam marks in all four.

The number line strips away the notation and leaves only the position. Two expressions that land at the same point are equal, full stop — even if one is a fraction and the other a percentage and the third a decimal.

Order these four numbers from smallest to largest: $\tfrac{3}{8}$, $0.4$, $35\%$, $\tfrac{2}{5}$

Step 1. Convert each to a decimal (easiest common ground).

\tfrac{3}{8} = 0.375 \qquad 0.4 = 0.4 \qquad 35\% = 0.35 \qquad \tfrac{2}{5} = 0.4

Why decimals as common ground: comparing decimals is just comparing digit-by-digit from the left, which is fast and reliable. Fractions with different denominators need a common denominator first; percentages need dividing by 100. Decimals are the cheapest format to compare in.

Step 2. Place on the line. 0.35 < 0.375 < 0.4, and 0.4 = 0.4, so \tfrac{2}{5} and 0.4 land on the same point.

Step 3. Write the order.

35\% \;<\; \tfrac{3}{8} \;<\; \tfrac{2}{5} = 0.4

A cleaner version of the question than it first looked — the decimal form revealed the hierarchy in a single sweep.

The tiny trap with percentages

A common slip: reading 28\% as "twenty-eight" and thinking it is larger than \tfrac{2}{7} because 28 > 2. Percentages are not the number 28 — they are the number \tfrac{28}{100} = 0.28. The percent sign means "per hundred," literally. If you strip the sign off, you have to divide by 100.

This is exactly the same trap as reading 0.28 as "twenty-eight" and thinking it is larger than 0.8 because 28 > 8. The digit count doesn't matter; the position on the line does. Drag the dots. The line never lies.