# Bar Models, Units and Ratio > Draw bars to compare amounts, share in a ratio, and solve "a fraction of one equals a fraction of another" problems. From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/bar-models/. Updated 3 November 2026. Year 11 and Year 12 boxes go deeper than the Year 10 text around them. ### Story time: Kim and her brother Kim and her brother saved 78 dollars together. Kim has 46 dollars less than 3 times what her brother has. How much does Kim have? Kim's teacher shows her a trick. Draw the brother's money as one bar. Kim's money is three of those bars, minus a little piece of 46 dollars. Together that is 4 bars minus 46 dollars, which equals 78 dollars. So 4 bars are worth $78 + 46 = 124$ dollars, one bar is 31 dollars, and Kim has $3 \times 31 - 46 = 47$ dollars. The picture made a hard problem easy. **In short:** A bar model draws each amount as a bar made of equal units, which turns a word problem into a picture. Find what 1 unit is worth, then multiply, and to share in a ratio, add the parts to find the total number of units. ## Before you start - Fractions of an amount (see the Fractions unit). - Equivalent fractions: making two fractions have the same numerator, like $\frac{3}{4} = \frac{6}{8}$ and $\frac{2}{5} = \frac{6}{15}$. - A **ratio** like $2 : 3$ compares two amounts: for every 2 of the first there are 3 of the second. ## The unitary method ### Key idea: Find one unit, then scale 1. Draw each amount as a bar made of equal **units**. 2. Use the facts to find what **1 unit** is worth. 3. Multiply to answer the question. ### Worked example: More, less and times Two whole numbers add to 600. The larger is 3 times the smaller plus 24. Smaller: 1 unit. Larger: 3 units $+ 24$. Together: 4 units $+ 24 = 600$. 4 units $= 576$, so 1 unit $= 144$. The larger number is $3 \times 144 + 24 = 456$. ## Sharing in a ratio ### Key idea: Add the parts To share an amount in the ratio $2 : 3$, there are $2 + 3 = 5$ parts. Find one part, then multiply. ### Worked example: Sharing marbles 120 marbles are shared between Alan and George in the ratio $2 : 3$. One part is $120 \div 5 = 24$. Alan gets 48 and George gets 72. ### Worked example: Three people 480 dollars is shared between Katie, James and Justin. Katie gets twice as much as James. Justin gets $\frac{1}{3}$ of what James gets. Give Justin 1 unit. Then James has 3 units and Katie has 6 units. Total 10 units $= 480$, so 1 unit $= 48$ and James gets $144$ dollars. ### Exam tip: Give the smallest share 1 unit Choosing the smallest amount as 1 unit avoids fractions of units. ## A fraction of one equals a fraction of another This type of question appears on almost every selective test. ### Key idea: Make the tops the same If $\frac{3}{4}$ of Effandy's money equals $\frac{2}{5}$ of Asdaly's money, rewrite both fractions with the **same numerator**: $\frac{3}{4} = \frac{6}{8}$ and $\frac{2}{5} = \frac{6}{15}$. So 6 parts out of Effandy's 8 equal 6 parts out of Asdaly's 15. Every part is the same size. **Effandy has 8 units and Asdaly has 15 units.** *Diagram: Bar model: Effandy's bar has 8 equal units and Asdaly's bar has 15 equal units of the same size* *All the units are the same size, so the two amounts can be compared directly.* ### Worked example: Then the story If Asdaly gives Effandy 35 dollars, they will have the same amount. Asdaly has $15 - 8 = 7$ units more. To make them equal, Asdaly gives away **half** the difference: $3\frac{1}{2}$ units $= 35$ dollars. So 1 unit $= 10$ dollars, and together they have $8 + 15 = 23$ units $= 230$ dollars. Effandy's money as a fraction of Asdaly's: $\frac{8}{15}$. ### Watch out: Giving halves the gap When one person gives money to another to make them equal, the amount given is **half** the difference. Giving the whole difference would just swap who is ahead. ### Challenge: Selective level: schools and savings $\frac{2}{3}$ of the students in School A equals $\frac{3}{5}$ of the students in School B. School B has 80 more students. Make the tops 6: $\frac{6}{9}$ and $\frac{6}{10}$. A has 9 units, B has 10 units. The difference of 1 unit is 80, so A has 720 and B has 800. $\frac{7}{10}$ of Delia's money equals $\frac{1}{5}$ of Sarah's. Make the tops 7: $\frac{7}{10}$ and $\frac{7}{35}$. Delia has 10 units and Sarah 35 units, which simplifies to $2 : 7$. Together they have 225 dollars, so 9 parts $= 225$, 1 part $= 25$ and Delia has 50 dollars. **Mind map** ```text Bar models Units Draw equal blocks Find 1 unit Multiply Ratio Add the parts Share by the parts Fraction of A = fraction of B Make the tops the same Bottoms give the units Giving to equal Give half the difference ``` ## Quick summary - Draw each amount as equal units, find 1 unit, then multiply. - To share in a ratio, add the parts to find how many parts in total. - Give the smallest share 1 unit to avoid fractions. - If a fraction of A equals a fraction of B, make the numerators equal. The denominators then give the units for A and B. - To make two amounts equal, the richer one gives away half the difference. ## Quiz Try these questions from the chapter quiz. 1. 120 marbles are shared in the ratio $2 : 3$. How many does the bigger share get? - A. 48 - B. 72 - C. 60 - D. 80 - Answer: B. 72. 5 parts, so 1 part is 24. The bigger share is $3 \times 24 = 72$. 2. 3 bananas are worth 2 oranges. How many oranges are 24 bananas worth? - A. 12 - B. 16 - C. 36 - D. 18 - Answer: B. 16. 24 bananas is 8 lots of 3 bananas, so 8 lots of 2 oranges: 16. 3. Sam has 45 postcards. Ben has 5 fewer than Sam. Bill has $1\frac{1}{4}$ times as many as Ben. How many do they have altogether? - A. 130 - B. 135 - C. 140 - D. 145 - Answer: B. 135. Ben 40, Bill $40 \times 1\frac{1}{4} = 50$. Total $45 + 40 + 50 = 135$. The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/bar-models/#quiz