The Bar Model Method

Updated 24 September 2026

A bar model is a drawing of the numbers in a word problem as rectangles, with equal amounts drawn the same length. Teachers in the United States call the same drawing a tape diagram, and in Singapore it is taught as the model method. The drawing turns the sentences of a question into lengths you can compare, and the calculation usually follows from the picture.

There are three shapes to learn: the part-whole model, the comparison model, and the before-and-after model. Fractions and percentages use the same bars, cut into equal parts.

What is a bar model?

A bar model is a rectangle drawn for each quantity in a problem, with equal amounts drawn as equal lengths. You label the quantity you know, mark the one you want with a question mark, and the drawing shows how the two are related. For example, 4 equal boxes hold 52 crayons altogether. Draw one bar for the 52 crayons and cut it into 4 equal parts. One part is 52 ÷ 4 = 13 crayons, so each box holds 13. The method works because the drawing replaces the words of the question with lengths, and lengths can be cut, compared and counted without guessing which operation the words are asking for. When the parts are not equal, for example one box holding 5 more crayons than each of the others, draw the extra 5 as a separate small block. Take it away first, then share what is left equally. Check the answer by putting it back into the story: 4 × 13 = 52.

How do you draw a part-whole model?

A part-whole model is one long bar for the whole amount, cut into the parts that make it up. When the parts are equal, cut the bar into the number of equal pieces that the denominator names. For example, of the 35 students in a class, $\frac{3}{5}$3/5 walk to school. Draw one bar for all 35 students and cut it into 5 equal parts. One part is 35 ÷ 5 = 7 students, so 3 parts are 3 × 7 = 21 students who walk. The model works because a fraction of an amount is a count of equal parts, and the drawing makes you find one part before you count them. When the question gives the part and asks for the whole, such as 21 students being $\frac{3}{5}$3/5 of a class, fill in the same bar the other way: 3 parts are 21, so 1 part is 7 and 5 parts are 35. Check: 21 walk and 14 do not, and 21 + 14 = 35.

How do you draw a comparison model?

A comparison model draws one bar for each quantity, lined up at the same starting edge, so that the difference between them shows as the piece of the longer bar that sticks out. For example, Leo and Maya share stamps in the ratio 3 : 5, and Maya has 48 more stamps than Leo. Draw Leo's bar as 3 units and Maya's as 5 equal units. The piece that sticks out is 2 units, and it stands for the 48 stamps, so 1 unit is 48 ÷ 2 = 24 stamps. Leo has 3 × 24 = 72 stamps and Maya has 5 × 24 = 120. The model works because the bars turn "3 : 5" and "48 more" into lengths you can line up. When the question gives the total instead of the difference, add the bars: 3 + 5 = 8 units. Check both facts: 72 and 120 divided by 24 give 3 and 5, and 120 − 72 = 48.

How do you solve before-and-after problems?

A before-and-after problem changes a quantity partway through the story. Draw one row of bars for before the change and one for after, and look for the quantity that stayed the same. For example, a jar holds red and blue beads in the ratio 2 : 5. After 12 red beads are added, the ratio becomes 4 : 5. The blue beads did not change, so blue is 5 units in both rows. Red went from 2 units to 4 units, so the 12 new beads are 2 units and 1 unit is 12 ÷ 2 = 6 beads. At first there were 12 red beads and 30 blue beads. The model works because drawing the unchanged quantity at the same length makes every unit mean one amount. When both quantities change but their total does not, keep the whole bar fixed instead. Check: after the change there are 24 red and 30 blue, and dividing both by 6 gives the ratio 4 : 5.

How does a bar model handle a fraction of the remainder?

When a question takes a fraction of what is left, cut the bar twice: first for the whole, then for the remainder. For example, Sam spends $\frac{1}{4}$1/4 of his money on a book and $\frac{2}{3}$2/3 of the rest on lunch, and he has 10 dollars left. Draw one bar and cut it into 4 equal parts. The book takes 1 part and 3 parts remain. Lunch takes $\frac{2}{3}$2/3 of the rest, which is 2 of those 3 parts. The last part is the 10 dollars left, so every part is 10 dollars and the whole bar is 4 × 10 = 40 dollars. The model works because it shows which whole each fraction belongs to, the step most often lost in words alone. When the second fraction does not fit the parts, cut every part into smaller equal pieces first. Check: $\frac{1}{4}$1/4 of 40 is 10, $\frac{2}{3}$2/3 of the remaining 30 is 20, and 40 − 10 − 20 = 10.

How does a bar model show a percentage?

A percentage bar is a bar that stands for 100%, cut into 10 or 100 equal parts. For example, a jacket costs 80 dollars and is sold at 15% off. Draw the bar for the full price and mark 15% of it as the discount. 10% of 80 dollars is 8 dollars and 5% is half of that, 4 dollars, so the discount is 8 + 4 = 12 dollars. The rest of the bar is 85% of the price: 80 − 12 = 68 dollars. The model works because 100% is always the whole bar, so any percentage is a share of a length you can see. When the question gives the price after the discount and asks for the original, read the bar the other way: 68 dollars fills 85 of the 100 parts, so one part is 68 ÷ 85 = 0.8 dollars and all 100 parts are 80 dollars. Check: 15% of 80 is 12, and 80 − 12 = 68.

When is algebra quicker than a bar model?

A bar model and an equation hold the same information: one unit of a bar is the letter of an equation, drawn as a length. For example, one number is 3 more than another, and together they make 17. The bar model draws two equal bars, adds a piece of 3 to one of them, and takes that 3 away from the total: 17 − 3 = 14 is two equal bars, so the smaller number is 7. Algebra writes the same steps as n + (n + 3) = 17, so 2n = 14 and n = 7. The drawing is quicker when the numbers are whole and the relationships are few, because it shows which step to take next. Algebra is quicker once there are several unknowns, fractions of unknowns, or quantities that cannot be drawn to scale, such as a changing speed. Check the answer in the question: 7 and 10 differ by 3, and 7 + 10 = 17.

The mistakes worth naming

• Drawing the bars by eye. If two quantities are in the ratio 3 : 5, their bars must be exactly 3 and 5 units long. A bar that is only roughly the right length hides the unit you are looking for.
• Taking a fraction of the wrong whole. In "$\frac{2}{3}$2/3 of the rest", the $\frac{2}{3}$2/3 belongs to the remainder, not to the whole bar. Cut the remainder, not the bar.
• Letting the unchanged quantity change length. In a before-and-after problem, the quantity that did not change must be drawn the same length in both rows.
• Dividing by the wrong number of units. Divide the known amount by the units it covers. In the stamp problem, 48 covers the 2 units that stick out, not all 8.

Where this leads next

The part-whole model is the picture behind fractions and percentages. The comparison and before-and-after models carry most of ratio, rates and proportion. Once a problem needs more than two or three unknowns, the same thinking moves into equations and inequalities, where each unit becomes a letter.

Practice it in the game

Math Challenge teaches bar models inside its lessons on fractions, percentages and ratio, starting with Bar Models for Ratio. Each set of applications below works its problems both ways: by the model method and by algebra.