090i - Problem Solving with Rates

Lesson 90 — Problem Solving with Rates

Strand: Measurement | Descriptor: AC9M8M05 | Duration: 45 minutes

Learning Intentions

• To apply rate calculation, conversion and comparison skills to solve varied practical problems.
• To use a rate to scale a quantity up or down proportionally.

Success Criteria

I can:

1. Choose the correct operation (multiply or divide) when using a rate to scale a quantity.
2. Solve multi-step problems that combine calculating, converting, and comparing rates.
3. Use a rate to convert between a real quantity and a scaled model or plan (e.g. recipes, exchange rates, medicine dosages).
4. Justify my working and check whether an answer is reasonable in context.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. Doubling a recipe’s quantity of an ingredient doubles the quantity of every other ingredient too, if the rate stays the same.
2. If you know a rate, you can find any unknown quantity connected to it by either multiplying or dividing.
3. A currency exchange rate tells you exactly how much you’ll receive with no fees or charges.
4. Scaling a rate up (e.g. from 1 serving to 4 servings) requires converting units first.

Answers: 1. Always — this is exactly what it means for two quantities to be in a constant rate/ratio relationship. 2. Always — a rate connects two quantities; you either scale up (multiply) or find “per 1” (divide). 3. Sometimes — real exchanges often include fees or a margin on top of the quoted rate. 4. Sometimes — only if the original rate and the target quantity are given in different units; if they already match, no conversion is needed.

Activities

Activity 1 — Fluency Review: Scaling with a Rate (12 min)

Explicit recap, then practice.

I do — scaling a recipe. A pancake recipe uses g of flour for every servings. How much flour is needed for servings?

I do — currency exchange. The exchange rate is AUD USD. Convert $250 AUD to USD.

We do: A paint tin covers per L. How much paint is needed to cover ?

You do: Solve each using the given rate:

1. A medicine dose is mg per kg of body mass. Find the dose for a kg child.
2. A printer produces pages per minute. How many pages in minutes?
3. An exchange rate is AUD GBP. Convert $420 AUD to GBP.
4. A concrete mix uses bags of cement per . How many bags are needed for ?

Answers: 1. mg; 2. pages; 3. ; 4. rate bags/, so bags (round up, as bags can’t be split).

Activity 2 — Applied Problems, including a Harder Multi-stage Rate Problem (21 min)

Pairs. Every answer must carry correct units and a one-sentence justification.

Problem 1. A factory’s machine produces bottles in hours. At the same rate, how long will it take to produce bottles?

Problem 2. A hiking group walks at a steady rate of km/h. If they walk for h min, how far do they travel?

Problem 3. A nurse must give a patient medication at a rate of mg per kg of body mass per day, split evenly across doses. For a kg patient, how many mg should be in each dose?

Problem 4 (harder). A wallpaper printing machine at Factory A prints wallpaper at per minute. A newer machine at Factory B prints per hours. Which machine is faster? An order for of wallpaper needs to be printed by the faster machine, running continuously — how many whole hours will it take, and how many extra minutes beyond that?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	First, compare two rates given in different units; then use the faster rate to solve a time problem.
What do you know?	Factory A: /min. Factory B: per hours — a different time unit.
Devise a plan	Convert Factory B’s rate to /min (or convert Factory A’s to /h) so they can be compared directly.
Carry out the plan	Factory B: /h. Factory A: /h.
Look back — which is faster?	Factory B (/h /h).
Devise a plan for the time question	time total area rate, using Factory B’s rate.
Carry out the plan	h h min h min.
Look back	Is hours reasonable for at roughly /h? , close to — consistent.

Answers: Problem 1 — rate bottles/h, time h. Problem 2 — h min h, distance km. Problem 3 — total daily dose mg, per dose mg. Problem 4 — Factory B is faster (/h vs /h); the order takes approximately hours and minutes.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A juice concentrate is mixed at a rate of part concentrate to parts water. How much water is needed for mL of concentrate?
2. A car uses fuel at L per km. How many litres are needed for a km trip?
3. Two labellers: Machine X labels jars per hour; Machine Y labels jars per hours. Which is faster?
4. Reasoning. A recipe scaling problem gives an answer of g of an ingredient. Explain why this might be a perfectly acceptable answer, even though you can’t measure “half a gram” precisely by eye.

Answers: 1. mL. 2. L. 3. Machine Y jars/h jars/h — Machine Y is faster. 4. Kitchen scales can typically measure to the nearest gram or half-gram, so g is a realistic, usable answer; rounding it further isn’t necessary unless the tool used genuinely can’t measure that precisely.

Common Misconceptions

Misconception	How to pre-empt it
Multiplying by the wrong quantity when scaling a rate (e.g. multiplying by the new time instead of the unit rate).	Always find the unit rate (“per 1”) explicitly as a first step before scaling to any other quantity.
Forgetting to convert mixed time units (e.g. “1 h 40 min”) into a single unit before multiplying by a rate.	Require every mixed time to be rewritten as a decimal or fraction of an hour (or all in minutes) before it is used in a rate calculation.
Rounding a “number of bags/tins/doses” answer down when the context requires rounding up (or vice versa).	Ask explicitly: “can you use a fraction of this item in real life?” before rounding, as in Activity 1 Q4.
Comparing two rates given in different time or mass units without converting first.	Model Problem 4 explicitly: always convert to a common unit before claiming one rate is faster or better.
Treating “hours” from a division as a decimal that can be read directly as hours and minutes (e.g. reading h as “21 hours 5 minutes”).	Show the conversion explicitly: min, not minutes.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A photocopier at a school library uses paper at a rate of reams per copies. If the library has reams left, how many copies can it make before running out?

Answer

E2 (AMC Junior style). A tap leaks at a constant rate, wasting L every minutes. At this rate, how many whole litres does it waste in a full -hour day?

Answer

108 litres.

E3 (Challenge). A currency converter charges no fee but rounds every converted amount down to the nearest cent. If the exchange rate is AUD USD, how many whole AUD would you need to convert to guarantee receiving at least USD, given the rounding-down rule?

Answer

Since amounts are rounded down, converting AUD would give slightly less than USD (), so at least 159 AUD is needed: USD ✓

Homework

1. A concrete mix uses bags of cement per . How many bags are needed for ?
2. A car travels at a constant L per km. How much fuel is needed for a km trip?
3. A juice recipe uses part cordial to parts water. How much cordial is needed to make L of juice in total?
4. Two pumps: Pump A empties a tank at L/min. Pump B empties L in minutes. Which pump is faster, and by how many L/min?
5. Reasoning. A scaling problem asks for the amount of an ingredient needed for a recipe. Explain why multiplying the original quantity by works, using the idea of a constant rate between ingredients.
6. Challenge. A tank is being drained at a rate of L per minutes. It starts with L. After how many whole minutes will the tank contain less than L for the first time?

Answers: 1. rate bags/; bags. 2. L. 3. L parts total, so part L of cordial. 4. Pump B L/min L/min — Pump B is faster, by L/min. 5. Every ingredient is connected to “1 recipe” by a fixed rate; scaling the whole recipe by scales every ingredient’s rate by the same factor, since all quantities stay in constant proportion to each other. 6. Drain rate L/min. Remaining , so after whole minutes the tank first contains less than L.