Mathematics · 01 / 23
Fractions: share the same whole
Middle school · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
A fraction compares a quantity with a whole divided into equal parts. Adding requires parts of the same size, so rewrite equivalent fractions first. Multiplication takes a fraction of another quantity. Division by a nonzero fraction multiplies by its reciprocal.
One whole in four equal parts. Three are selected.
- 01Same whole
- 02Equal parts
- 03Common denominator
- 04Add the parts
WORKED EXAMPLE
From principle to reasoning
For 3/4 + 1/2, rewrite 1/2 as 2/4. Three quarters plus two quarters gives 5/4: one whole and one quarter.
Division with concrete parts
Three wholes contain six halves: 3 ÷ (1/2) = 3 × 2. Three quarters measured in halves gives 3/4 × 2 = 3/2. That result counts half-sized parts; it does not make three quarters greater than one whole.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Calculate 2/3 + 1/6.
One hint
Look for sixths.
Reveal and understand the solution
2/3 = 4/6, so 4/6 + 1/6 = 5/6. Five of six parts is less than one whole.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Why is 1/2 + 1/3 not 2/5?
A hint to get started
The parts must have equal size.
Step-by-step solution
- One half is 3/6 and one third is 2/6.
- Add equal-sized parts: 5/6.
- 2/5 is below 1/2, so it also fails a magnitude check.
Which step did you miss? Retry without help, then explain why the method works.
Calculate and simplify 3/4 ÷ 2/5.
A hint to get started
Divide by a nonzero fraction by multiplying by its reciprocal.
Step-by-step solution
- Write 3/4 × 5/2.
- Multiply to get the irreducible fraction 15/8.
- Dividing by 0.4 does increase the original number.
Which step did you miss? Retry without help, then explain why the method works.
A tank is 3/5 full. A quarter of that water is removed. What fraction remains full?
A hint to get started
The quarter refers to the water, not the whole capacity.
Step-by-step solution
- Three quarters of the original water remain.
- Multiply 3/5 by 3/4: 9/20.
- The removed amount is 3/20; 12/20 − 3/20 checks the result.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Adding denominators changes the size of the parts; it is not the addition rule.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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