Mathematics • Year 10 • Unit 2 • Lesson 9
Fraction Equations in the Real World
Use LCDs and cross-multiplication on real-world setups — sharing pizza, scaling maps, fuel-economy rates and average speed. Then defend why a restriction matters when the unknown appears in a denominator.
1. Word problems
For each problem: define the variable, set up the fraction equation, solve it, and state the answer in context.
1.1 — Sharing a pizza. A group of friends shares a pizza equally. Mai eats one quarter of the pizza, Sam eats one third, and 5 slices are left over. The whole pizza has x slices.
(a) Write expressions for the number of slices Mai and Sam each ate.
(b) Set up an equation that says Mai's slices + Sam's slices + 5 = total slices.
(c) Clear fractions using the LCD and solve for x. 4 marks
1.2 — Map scale. On a hiking map, 2 cm represents 5 km of real distance. Two campsites are x cm apart on the map and 14 km apart in real life.
(a) Write a proportion: x/14 = 2/5.
(b) Cross-multiply and solve for x. State the restriction (if any). 3 marks
1.3 — Fuel economy. A car uses 1 litre of fuel for every x km it drives. On a long trip the car uses 24 L to drive 360 km.
(a) Write the equation 1/x = 24/360. State the restriction.
(b) Cross-multiply and solve for x. Interpret the answer (km per litre). 3 marks
1.4 — Average speed. Mai cycles to school at 18 km/h and walks home at 6 km/h. The total distance each way is d km. Her total travel time is 1 hour.
(a) Write expressions for the time taken to cycle and to walk (use time = distance/speed).
(b) Set up the equation total time = 1, then solve for d. 4 marks
1.5 — Concentration of juice. A bottle contains 5 mL of cordial in x mL of water. The recipe says 1 mL cordial per 4 mL water. Find x so the bottle matches the recipe.
(a) Write the proportion 5/x = 1/4. State the restriction.
(b) Solve and state x in mL. 3 marks
2. Explain your thinking
Communication, not just numbers. 4 marks
2.1 A classmate solves 6/(x − 3) = 2 and gets the answer x = 3. They are very pleased because the numbers feel clean. Using the lesson rules for restrictions and algebraic fractions, explain (i) why x = 3 cannot be a valid solution, (ii) what the actual restriction is for this equation, and (iii) how to find the real solution. Show the corrected working.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pizza
(a) Mai = x/4, Sam = x/3. (b) x/4 + x/3 + 5 = x. (c) Multiply every term by 12: 3x + 4x + 60 = 12x → 7x + 60 = 12x → 60 = 5x → x = 12 slices. Check: 12/4 + 12/3 + 5 = 3 + 4 + 5 = 12 ✓.
1.2 — Map scale
(a) x/14 = 2/5 (no restriction needed since both denominators are constants). (b) Cross-multiply: 5x = 28 → x = 5.6 cm.
1.3 — Fuel economy
(a) Restriction: x ≠ 0. (b) Cross-multiply: 360 = 24x → x = 15. The car gets 15 km per litre.
1.4 — Average speed
(a) Cycle time = d/18 h; walk time = d/6 h. (b) d/18 + d/6 = 1. LCD = 18: d + 3d = 18 → 4d = 18 → d = 4.5 km. Check: 4.5/18 + 4.5/6 = 0.25 + 0.75 = 1 h ✓.
1.5 — Cordial
(a) 5/x = 1/4 with x ≠ 0. (b) Cross-multiply: 20 = x → x = 20 mL of water.
2.1 — Explain (sample response)
(i) When x = 3, the denominator x − 3 becomes 0, and division by 0 is undefined — so the original equation has no meaning at x = 3. It cannot be a valid solution no matter what the rest of the algebra gives. (ii) The restriction for this equation is x ≠ 3. (iii) Correctly solve by multiplying both sides by (x − 3): 6 = 2(x − 3) → 6 = 2x − 6 → 12 = 2x → x = 6. Check x ≠ 3 ✓. Verify: 6/(6 − 3) = 6/3 = 2 ✓.
Marking: 1 mark for explaining division by zero; 1 mark for the restriction x ≠ 3; 1 mark for correct algebra to x = 6; 1 mark for the substitution check.