Mastering "Sum and Difference" Word Problems Using the Unit Method
Why can't we just divide the sum in half right away?
In a Sum & Difference problem, the two items are unequal. We cannot divide by 2 because one item has more than the other!
To solve it, we must first make both bars identical in length. There are two ways to do this:
Subtract the difference ($D$) from the total sum ($S$). This leaves 2 equal smaller units!
Add the difference ($D$) to the total sum ($S$). This gives 2 equal larger units!
Memorize this checklist for every test problem!
Underline the Sum (total) and the Difference (how many more/fewer). Identify who has more.
Draw two bars starting from the same left baseline. The longer bar has a dashed box at the end labeled with the difference.
Deduct the difference from the total sum:
Total − Difference = Equal Units
Divide to find 1 unit. Re-read the question: did it ask for the smaller item, larger item, or something in the future?
Every Sum and Difference question is a variation of these 5 patterns:
Model Concept: $B$ has 80 more than $A$. If you remove 80 from 540, you are left with two of $A$'s units.
Model Concept: Relate everything to the smallest quantity (Class 4A = 1 unit). Class 4B = $1\text{u} + 35$. Class 4C = $1\text{u} + 35 + 50 = 1\text{u} + 85$.
Model Trap: Perimeter $= 2\text{ lengths} + 2\text{ widths} = 180\text{ m}$. You must first divide by 2 to find 1 length + 1 width = 90 m!
Conservation of Total Principle: When items pass between two people, the Total is unchanged!
Replacement Model: Treat 1 basketball as 1 unit. Soccer ball is $1\text{ unit} + \$15$. Total is $3\text{ units} + \$15$.
Change the Sum and Difference to see the bar model update live!
Watch out for these common test mistakes!
Mistake: Doing $540 \div 2 = 270$ first, then adding 80.
✔️ Fix: Always subtract the extra difference before dividing by 2: $(540 - 80) \div 2 = 230$!
Mistake: Finding 1 unit = 230, and writing 230 as the answer when the question asked: "How much did Emma have?"
✔️ Fix: Re-read the last sentence! Emma had $230 + 80 = 310$.
Mistake: Treating Perimeter (180 m) as Length + Width.
✔️ Fix: Perimeter $= 2 \times (\text{Length} + \text{Width})$. Halve it first: $180 \div 2 = 90\text{ m}$!
Mistake: Thinking Nathan had 45 more than Oliver at first because he gave 45.
✔️ Fix: Giving 45 means the initial gap was $2 \times 45 = 90$! Always work backwards from the equal state.
Click any card to flip and reveal the bar model working!
Sam and Max have 84 marbles in total. Sam has 16 more than Max. How many does Max have?
A rectangle has a perimeter of 60 cm. Its length is 6 cm longer than its width. What is the length?
Leo and Maya have 100 stickers. Leo gives 15 stickers to Maya and they become equal. How many did Leo have at first?
Take the official 10-item Singapore Bar Model Quiz with interactive diagrams, immediate feedback, and scoring!