MAT8 Q2W8D3: Strategies for Solving Financial Problems

🎯 Learning Goals

By the end of the lesson, you will be able to:

1. Solve multi-step financial problems (best buy, surplus or deficit, profit or loss) and show correct computations in MathML - accurately answering at least 8 of 10 practice items.
2. Decide between options (cash vs installment with fees, commuter pass thresholds, break-even quantities) and justify your choice using a numeric criterion (unit rate, total cost, or margin), achieving ≥ 80% decision accuracy.
3. Construct a short solution explanation that states the goal, constraints, and computed values (using MathML like unit rate, total cost, or simple interest) in three concise sentences for 3 of 3 given scenarios.

🧩 Key Ideas & Terms

• Multi-step problem - a situation requiring two or more linked computations (for example, compute unit rate, then compare totals).
• Constraint - a limit (cash on hand, time, storage) that affects the feasible choice.
• Decision criterion - the computed value used to compare options (for example, unit rate, total cost, or profit).
• Unit rate / Cost per unit - price per one unit of measure.
  $\text{Cost per unit}=\frac{\mathrm{Price}}{\mathrm{Quantity}}$
• Total cost - the sum of all costs (fixed + variable + fees).
  $\text{Total cost}=\text{Fixed}+\text{Variable}+\text{Fees}$
• Revenue - income from selling items.
  $\text{Revenue}=\text{Price}\times \text{Quantity sold}$
• Profit / Loss - the difference between revenue and total cost.
  $\mathrm{Profit}=\mathrm{Revenue}-\mathrm{Cost}$
  $\mathrm{Loss}=\mathrm{Cost}-\mathrm{Revenue}$
• Surplus / Deficit - difference between income and expenses.
  $\mathrm{Surplus}=\mathrm{Income}-\mathrm{Expenses}$
  $\mathrm{Deficit}=\mathrm{Expenses}-\mathrm{Income}$
• Fixed cost - a cost that does not change with quantity (for example, stall or setup fee).
• Variable cost - a cost that changes with quantity produced or sold (for example, materials per item).
• Margin (per unit) - selling price minus variable cost per unit.
  $\text{Margin}=\text{Price}-\text{Variable cost}$
• Break-even point - quantity where profit becomes zero.
  $\text{Break-even Q}=\frac{\text{Fixed cost}}{\text{Margin per unit}}$
• Threshold (pass vs pay-per-use) - minimum usage where a flat pass becomes cheaper.
  $\text{Pass cost}\le \text{Rate}\times R$
• Installment - paying in parts over time; total may increase due to fees or interest.
  $\text{Base payment}=\frac{\text{Cash price}}{\text{Number of periods}}$
• Service or processing fee - an added amount per period that increases total installment cost.
• Simple interest - interest computed on principal only.
  $I=P\times r\times t$
• Average balance (estimate) - an approximate mean amount held over time for quick interest estimates when deposits are added periodically.
• Opportunity cost - the value of the best alternative you give up when you choose something.

🔄 Quick Recall / Prior Knowledge

1) Decision criterion - The instruction says: "Find which bottle is cheapest per liter." What criterion is being used

Show Answer

Unit rate or cost per unit. You compute a price per 1 liter to compare options fairly.

2) Quick unit-rate compare (rice) - Option A: 2 kg for ₱130. Option B: 5 kg for ₱320. Compute ₱ per kg and pick the cheaper per kilogram.

Show Answer

A:

$\text{₱/kg}=\frac{130}{2}=65$

B:

$\text{₱/kg}=\frac{320}{5}=64$

Cheaper per kg: Option B at ₱64/kg.

3) Surplus or deficit - Weekly income ₱1,500; weekly expenses ₱1,450. Is there a surplus or deficit, and how much

Show Answer

$\mathrm{Surplus}=1500-1450=50$

₱50 surplus.

4) Profit quick check with fixed and variable costs - You sell notebooks at ₱25. Variable cost is ₱15 each. Fixed stall fee is ₱200. If you sell 30 units, find revenue, total cost, and profit.

Show Answer

Revenue:

$30\times 25=750$

Variable cost:

$30\times 15=450$

Total cost:

$450+200=650$

Profit:

$750-650=100$

Profit = ₱100.

5) Pass threshold inequality - Fare is ₱18 per ride. A monthly pass costs ₱720. Find the minimum rides R so that the pass is cheaper than or equal to paying per ride.

Show Answer

$720\le 18\times R$

$R\ge \frac{720}{18}=40$

At least 40 rides.

6) Break-even refresher - Selling price ₱50, variable cost ₱32, fixed cost ₱900. Find margin per unit and break-even quantity.

Show Answer

Margin:

$\text{Margin}=50-32=18$

Break-even quantity:

$\text{BEQ}=\frac{900}{18}=50$

50 units to break even.

7) Installment quick math - Cash price ₱2,400. Installment: 3 months, equal base payment, plus fee ₱40 per month. Find base payment, monthly outlay, total paid, and extra cost vs cash.

Show Answer

Base payment:

$\frac{2400}{3}=800$

Monthly outlay:

$800+40=840$

Total paid:

$840\times 3=2520$

Extra vs cash:

$2520-2400=120$

₱120 extra due to fees.

8) Simple interest quick estimate - Deposit ₱5,000 at 3 percent per year for 9 months. Compute simple interest.

Show Answer

Time in years:

$t=\frac{9}{12}=0.75$

Interest:

$I=P\times r\times t=5000\times 0.03\times 0.75=112.5$

₱112.50 interest.

📖 Explore the Lesson

Welcome to the problem solving day. Yesterday you practiced describing financial situations using lenses like best buy, profit and loss, installment, and surplus or deficit. Today you will solve multi step problems that use those lenses to reach decisions. You will also explain why your answers make sense, using small MathML computations that anyone in class can follow. Read each part slowly, respond to the guiding questions, and open each hidden answer only after you have tried. Mini summaries after checkpoints will help you keep track of what you know so far.

Part A - From description to decision

A good financial solution starts from a clear description. You have already learned to name the lens and list the important numbers. Now you will move from description to decision by applying a small decision rule. Here are the most common rules you will use today.

1. Best buy rule - pick the option with the lower unit rate.
   $\text{Unit rate}=\frac{\text{Price}}{\text{Quantity}}$
2. Surplus or deficit rule - compare income and expenses.
   $\mathrm{Surplus}=I-E,\mathrm{Deficit}=E-I$
3. Profit and loss rule - compute revenue and total cost first, then subtract.
   $\mathrm{Revenue}=p\times q,\mathrm{Cost}=\text{Fixed}+\text{Variable},\mathrm{Profit}=\mathrm{Revenue}-\mathrm{Cost}$
4. Installment rule - find base payment, add fees, multiply by number of periods, then compare with cash.
   $\text{Base payment}=\frac{\text{Cash price}}{\text{Number of periods}}$
5. Threshold rule for passes or bundles - solve the inequality where a flat price becomes cheaper than paying per use.
   $\text{Pass cost}\le \text{Rate}\times R$

Checkpoint A1 - Identify your rule

Match each short description to a decision rule.

1. Compare two cereal bags by pesos per kilogram.
2. Decide if your weekly cash covers transport and food.
3. Compute how many stickers you must sell to cover a stall fee.
4. Decide if a month pass is cheaper than paying per ride.

Show Answer

1) Best buy rule. 2) Surplus or deficit rule. 3) Profit and loss rule with break even. 4) Threshold rule for passes.

Mini summary - A decision rule turns a description into a solution. You will practice choosing the correct rule before computing.

Part B - Best buy problems that feel real

Best buy is more than dividing price by quantity. In real stores you meet different units, sizes, and sometimes limits that affect what you can actually buy. You will handle three real twists today: unit conversion, wastage or unusable amounts, and cash or storage limits.

B1 - Unit conversion in groceries

You want tomato sauce for a weekend cook off. The brand has a 400 g can and a 0.9 kg pouch. Prices are below.

• 400 g can at ₱42
• 0.9 kg pouch at ₱84

Guiding question - how do you get both to kilograms or both to grams so you can compare fairly

Show Answer

Convert both to kilograms or grams. 400 g = 0.4 kg. Cost per kg for the can:

$\text{₱/kg}=\frac{42}{0.4}=105$

Pouch 0.9 kg:

$\text{₱/kg}=\frac{84}{0.9}=93.33$

The pouch is the better buy per kg.

Guiding question - if your recipe needs only 600 g, which option is cheaper for this one meal if leftovers may spoil

Show Answer

To reach at least 600 g you need either two cans or one pouch. Two cans give 800 g for ₱84. One pouch gives 900 g for ₱84. Both are ₱84. If leftovers spoil fully, the option that gives exactly what you need wastes less. Two cans leave 200 g, one pouch leaves 300 g. Two cans waste less for the same price in this single meal situation.

Mini summary - Convert units first. Then check the quantity you actually need because wastage can change the decision.

B2 - Wastage in a school art project

Your class buys glue for a poster project. Option A is a 200 ml bottle at ₱60. Option B is a 1 liter jug at ₱270. You estimate each team uses 180 ml and there are 5 teams. You also know that once opened, leftover glue dries out after two days, so it cannot be used later.

Guiding question - if 5 teams need 180 ml each, how much glue is required

Show Answer

$5\times 180=900$

900 ml or 0.9 liter.

Guiding question - how many of each option must you buy to reach at least 900 ml

Show Answer

Option A bottles required:

$\frac{900}{200}=4.5$

Buy 5 bottles. Cost:

$5\times 60=300$

Option B jugs required: one jug of 1 liter is enough. Cost ₱270.

Guiding question - which option wastes less glue on the event day

Show Answer

Five bottles give 1,000 ml. Waste is 1,000 − 900 = 100 ml. One jug gives 1,000 ml. Waste is also 100 ml. Both waste the same amount if used the same way. The jug is cheaper overall at ₱270 vs ₱300, so it is the better buy here.

Mini summary - Include wastage and minimum quantities in your comparison. Unit rate is not the only factor.

B3 - Cash or storage limits

A family has only ₱500 today and limited storage. Rice options are 5 kg at ₱260 or 10 kg at ₱500. Unit rate says 10 kg is cheaper per kg. How does cash and space affect the decision

Guiding question - what can the family afford today and store safely

Show Answer

Their cash exactly equals the 10 kg price. If storage is safe and dry, they can purchase 10 kg and benefit from the lower unit cost. If storage is not safe or they must save some cash for transport or vegetables, the 5 kg may be the practical choice despite the higher unit rate.

Mini summary - A complete best buy solution states the unit rate decision and also checks cash and storage limits.

Part C - Surplus or deficit with goal tracking

Solving for surplus or deficit is straightforward. Real families track goals over weeks and months. You will solve with time frames, categories, and small adjustments.

C1 - Weekly plan with a goal

Allowance per week is ₱800. The plan is food ₱420, transport ₱220, and save ₱180.

Guiding question - is the plan feasible as written

Show Answer

Sum of planned items:

$420+220+180=820$

Allowance is ₱800, so there is a deficit of ₱20. The plan is not feasible as written.

Guiding question - adjust only one category to remove the deficit while keeping some savings. Suggest one fix and show the number.

Show Answer

Reduce food by ₱20 to ₱400. New total:

$400+220+180=800$

Now the plan balances and still saves ₱180.

Mini summary - A small numeric adjustment can restore balance. Always show your arithmetic.

C2 - One time shock inside a monthly plan

A parent expects ₱10,000 for the month. Planned expenses are ₱3,800 food, ₱1,400 transport, ₱1,600 utilities, and ₱1,200 school needs. The rest goes to savings. In week 3 a medical bill of ₱1,500 appears.

Guiding question - compute the surplus or deficit after including the medical bill

Show Answer

Planned expenses before shock:

$3800+1400+1600+1200=8000$

With medical bill:

$8000+1500=9500$

Surplus is:

$\mathrm{Surplus}=10000-9500=500$

Guiding question - give two actions that protect the plan for the rest of the month

Show Answer

1) Use the remaining ₱500 as partial buffer and reduce optional spending for the last week. 2) Choose best buy options for food items to stretch pesos per kilogram. The goal is to avoid a deficit by the end of the month.

Mini summary - When shocks occur, recompute totals and adjust categories. You can still end the month with a surplus.

Part D - Profit and loss with break even and target profit

This part uses a small venture to connect fixed cost, variable cost, price, and quantity. You will solve for profit at a given quantity and also solve backward for the quantity needed to hit a goal.

D1 - Start with a simple booth

You sell school pins. Price per pin is ₱30. Variable cost per pin is ₱14. One time table fee is ₱400.

Guiding question - write the formulas for revenue, variable cost, total cost, and profit using quantity q

Show Answer

$R=30\times q$

$\mathrm{VC}=14\times q$

$\mathrm{TC}=400+\mathrm{VC}$

$P=R-\mathrm{TC}$

Guiding question - compute the break even quantity using margin per unit

Show Answer

Margin per pin:

$\text{Margin}=30-14=16$

Break even quantity:

$\text{BEQ}=\frac{400}{16}=25$

You must sell 25 pins to break even.

Guiding question - if you expect to sell 40 pins, what profit will you make

Show Answer

Revenue at 40:

$30\times 40=1200$

Variable cost at 40:

$14\times 40=560$

Total cost:

$400+560=960$

Profit:

$1200-960=240$

Mini summary - Profit comes from margin multiplied by quantity minus fixed cost. Break even uses fixed cost divided by margin.

D2 - Solve backward for a target profit

Same pin booth. You want a target profit of ₱400. How many must you sell

Guiding question - write the profit equation and solve for q

Show Answer

$P=30\times q-(400+14\times q)$

$P=(16\times q)-400$

$400=16\times q-400$

$800=16\times q$

$q=\frac{800}{16}=50$

You must sell 50 pins to earn ₱400.

Mini summary - A target profit turns the profit formula into an equation you can solve for quantity.

D3 - Sports fundraiser example with two supplier plans

A sports club wants team shirts. Supplier A price is ₱240 per shirt, no setup fee. Supplier B price is ₱198 per shirt with a ₱1,200 one time setup. The club estimates they can sell between 40 and 80 shirts at ₱300 each. Which supplier gives higher profit for a given quantity

Guiding question - write profit functions for each supplier in terms of quantity q sold at the club price of ₱300

Show Answer

For Supplier A: revenue is 300q. Cost is 240q. Profit_A:

$P_A=300\times q-240\times q=60\times q$

For Supplier B: revenue is 300q. Cost is 198q + 1200. Profit_B:

$P_B=300\times q-(198\times q+1200)=102\times q-1200$

Guiding question - for which quantities does Supplier B yield higher profit than Supplier A

Show Answer

Compare 102q − 1200 versus 60q. Solve 102q − 1200 ≥ 60q. Subtract 60q:

$42\times q\ge 1200$

Divide by 42:

$q\ge \frac{1200}{42}\approx 28.57$

For 29 or more shirts, Supplier B produces higher profit. The club expects between 40 and 80, so Supplier B is the better choice in that range.

Mini summary - With competing plans, write profit in terms of q, then compare the expressions to find the crossover quantity.

Part E - Installments and save then buy comparisons

E1 - Phone on terms

Cash price is ₱9,600. Installment plan is 6 months, equal base payment, plus processing fee ₱75 per month. Another approach is to save ₱1,600 per month for 6 months and buy in cash.

Guiding question - compute base payment, monthly outlay, and total paid under installment

Show Answer

Base payment:

$\frac{9600}{6}=1600$

Monthly outlay including fee:

$1600+75=1675$

Total paid in 6 months:

$1675\times 6=10050$

Guiding question - compare installment total cost with save then buy total

Show Answer

Save then buy total is the cash price of ₱9,600. Installment total is ₱10,050. Extra cost of installment is:

$10050-9600=450$

Guiding question - give one reason a family might still choose installment

Show Answer

They may need the phone immediately for work or school and do not have the cash today. The fee is the cost of early access.

Mini summary - Always compute monthly outlay and total cost. Compare with cash to measure the cost of paying over time.

E2 - Handling fee on remaining balance (month by month view)

A shop offers a laptop with a 12 month plan. Cash price is ₱24,000. Base share ignores fees and is evenly split. The shop also adds a handling fee of 2 percent of the remaining balance each month. You will compute the first two months.

Guiding question - compute the base share and month one fee

Show Answer

Base share without fees:

$\frac{24000}{12}=2000$

Month one fee:

$0.02\times 24000=480$

Month one payment is 2000 + 480 = 2480.

Guiding question - compute the remaining balance after month one and the month two fee

Show Answer

Remaining balance after base share:

$24000-2000=22000$

Month two fee:

$0.02\times 22000=440$

Month two payment is 2000 + 440 = 2440.

Mini summary - A fee on the remaining balance shrinks over time as the balance falls. This makes early months more expensive.

Part F - Threshold decisions for passes and bundles

F1 - Commuter pass

Single ride costs ₱17. A monthly pass costs ₱700 and covers up to 50 rides. You will find the minimum number of rides R where the pass is cheaper.

Guiding question - set up and solve the inequality

Show Answer

Pass is cheaper when pass cost is less than or equal to pay per ride:

$700\le 17\times R$

Divide both sides by 17:

$R\ge \frac{700}{17}\approx 41.18$

The smallest whole number that satisfies this is 42 rides.

Guiding question - if you commute 22 school days with 2 rides per school day, does the pass save money this month

Show Answer

Your rides are:

$22\times 2=44$

Since 44 is at least 42, the pass is the cheaper choice this month.

Mini summary - The threshold is found by dividing pass price by per use rate, then rounding up to the next whole use.

F2 - Sports streaming package threshold

A pay per match rate is ₱55. A monthly pass costs ₱500 for unlimited matches. Find the minimum number of matches m where the pass is cheaper.

Guiding question - write and solve the inequality

Show Answer

Set pass ≤ pay per match:

$500\le 55\times m$

Solve:

$m\ge \frac{500}{55}\approx 9.09$

The smallest whole number is 10 matches.

Mini summary - A pass is useful only if your usage crosses a clear numeric threshold.

Part G - Design, construction, and navigation contexts

G1 - Construction mix choice

A small construction team must buy cement mix. Store X offers 20 kg bags for ₱190. Store Y offers 25 kg bags for ₱230. Delivery fee is ₱300 regardless of how many bags you buy. The team needs at least 200 kg for a weekend project.

Guiding question - compute cost per kilogram for each store and then total cost for exactly 200 kg if you round up to full bags

Show Answer

Unit rate X:

$\text{₱/kg}=\frac{190}{20}=9.5$

Unit rate Y:

$\text{₱/kg}=\frac{230}{25}=9.2$

Bags needed for 200 kg:

$\frac{200}{20}=10$

$\frac{200}{25}=8$

Goods cost X: 10 × 190 = ₱1900. Goods cost Y: 8 × 230 = ₱1840. Add delivery ₱300. Total X = ₱2200, Total Y = ₱2140. Store Y is cheaper.

Guiding question - if the team actually needs 220 kg, recompute the totals

Show Answer

X bags for 220 kg:

$\frac{220}{20}=11$

Y bags for 220 kg:

$\frac{220}{25}=8.8$

Round up to 9. Goods cost X: 11 × 190 = ₱2090. Goods cost Y: 9 × 230 = ₱2070. Add delivery ₱300. Total X = ₱2390. Total Y = ₱2370. Store Y still wins, but by a small margin.

Mini summary - In construction, the better unit rate often stays better after rounding up to whole bags, but always check totals including delivery.

G2 - Sports club water plan

A sports club must supply water for a weekend. Option A is 500 ml bottles at ₱9 each. Option B is 5 gallon jugs with a dispenser. One gallon is 3.785 liters. A 5 gallon jug costs ₱180 and the dispenser rental is ₱250. The club expects 120 athletes, each to drink around 1.5 liters per day for 2 days.

Guiding question - estimate total liters needed

Show Answer

Per athlete:

$1.5\times 2=3$

For 120 athletes:

$120\times 3=360$

Liters total.

Guiding question - compute total cost for Option A using 500 ml bottles

Show Answer

Each bottle is 0.5 liter. Bottles needed:

$\frac{360}{0.5}=720$

Cost:

$720\times 9=6480$

Guiding question - compute total cost for Option B using 5 gallon jugs and a single dispenser

Show Answer

One 5 gallon jug is:

$5\times 3.785=18.925$

Liters. Jugs needed:

$\frac{360}{18.925}\approx 19.01$

Round up to 20 jugs. Jug cost:

$20\times 180=3600$

Add dispenser ₱250. Total is ₱3850. Option B is far cheaper for this volume.

Mini summary - For large volumes, bulk containers plus a small rental can beat many single use bottles.

G3 - Navigation and fuel decision

A family will drive 300 kilometers for a reunion. Car A uses 10 km per liter. Car B uses 13 km per liter. Gasoline price is ₱70 per liter. Car B rental fee is ₱500 higher than Car A. Which car gives the lower total trip cost

Guiding question - compute fuel liters needed for each car

Show Answer

For Car A:

$\text{Liters}=\frac{300}{10}=30$

For Car B:

$\text{Liters}=\frac{300}{13}\approx 23.08$

Guiding question - compute fuel cost for each and then add the rental difference

Show Answer

Fuel cost A:

$30\times 70=2100$

Fuel cost B:

$23.08\times 70\approx 1615.6$

Rental difference is ₱500 more for B. Total A = base rental R + 2100. Total B = (R + 500) + 1615.6 = R + 2115.6. Car A is cheaper by about ₱15.6 in this setup. If the trip were longer, the higher efficiency could overtake the rental difference.

Mini summary - Some choices depend on distance. A longer trip can change which option is better.

Part H - Multi step case studies with full explanations

H1 - Community pantry plan

A community pantry has ₱4,500 to refill rice and cooking oil. Options are: Rice 5 kg at ₱260 or 10 kg at ₱500. Oil 900 ml at ₱132 or 1.8 L at ₱250. They want at least 30 kg of rice and at least 9 liters of oil. Minimize total cost. Assume storage is safe and dry.

Guiding question - what combination of rice packs reaches 30 kg at minimum cost

Show Answer

Unit rate rice 5 kg is 52 per kg. Unit rate rice 10 kg is 50 per kg. To hit at least 30 kg, buy three 10 kg sacks for 30 kg at 3 × 500 = ₱1500. This is cheaper than mixing six 5 kg packs at ₱1560.

Guiding question - what combination of oil packs reaches at least 9 liters at minimum cost

Show Answer

Unit rate 0.9 L is about 146.67 per liter. Unit rate 1.8 L is about 138.89 per liter. To reach at least 9 liters, buy five 1.8 L jugs which give 9 liters exactly for 5 × ₱250 = ₱1250. Using 0.9 L packs needs 10 packs at ₱1320, which is more expensive.

Guiding question - compute total cost and verify it fits the budget

Show Answer

Rice total ₱1500 plus oil total ₱1250 equals ₱2750. This fits the ₱4500 budget and leaves ₱1750 for vegetables or noodles. The solution meets both minimums at lowest cost among the given choices.

Three sentence explanation model: 1) Goal - minimize cost while reaching 30 kg rice and 9 L oil. 2) Numbers - pick 3 sacks of 10 kg at ₱500 each and 5 jugs of 1.8 L at ₱250 each. 3) Decision with computation - total ₱2750 which fits the ₱4500 budget and leaves ₱1750 for other goods.

Mini summary - Structure your explanation with goal, numbers, and computation driven decision.

H2 - School print shop break even and pricing

A class plans a print shop to fund a field trip. Each booklet uses paper and ink that cost ₱22 per booklet, and they will rent a stapler for ₱200 for the day. They want at least ₱1,000 profit for the trip. What price should they set if they expect to sell 120 booklets

Guiding question - write the profit formula in terms of price p and quantity q = 120

Show Answer

Revenue is p × 120. Variable cost is 22 × 120. Total cost is 200 + 22 × 120. Profit is:

$P=p\times 120-(200+22\times 120)$

Guiding question - set P = 1000 and solve for p

Show Answer

Compute 22 × 120 = 2640. Cost = 200 + 2640 = 2840.

$1000=p\times 120-2840$

Add 2840:

$3840=p\times 120$

Divide by 120:

$p=\frac{3840}{120}=32$

They must price at least ₱32 per booklet to hit ₱1,000 profit if 120 copies sell.

Three sentence explanation model: 1) Goal - earn ₱1,000 profit at 120 copies. 2) Numbers - variable cost ₱22 each, fixed cost ₱200, so total cost is 2840. 3) Decision - set price to at least ₱32 since 32 × 120 − 2840 = 1000.

Mini summary - When price is unknown, fix the target profit and quantity, then solve the equation for price.

H3 - Design choice with printing sizes

A student council wants posters. Two printers offer different sizes and prices. Printer A prints large A2 posters for ₱80 each with minimum 30. Printer B prints A3 posters for ₱48 each with minimum 40. The council has a wall area limit and wants at least 2 square meters of total poster area but wants to minimize cost. Assume approximate areas. A2 is about 0.25 square meter. A3 is about 0.125 square meter.

Guiding question - compute total area at each minimum order and decide if the minimum meets the area requirement

Show Answer

A2 minimum area:

$30\times 0.25=7.5$

A3 minimum area:

$40\times 0.125=5$

Both exceed the 2 square meter requirement.

Guiding question - compute total cost at each minimum and pick the cheaper plan that still covers the area requirement

Show Answer

Cost A at minimum:

$30\times 80=2400$

Cost B at minimum:

$40\times 48=1920$

Printer B is cheaper and still meets the area need.

Three sentence explanation model: 1) Goal - reach at least 2 square meters with minimum spending. 2) Numbers - A2 gives 7.5 sq m for ₱2400, A3 gives 5 sq m for ₱1920. 3) Decision - choose Printer B to save ₱480 while exceeding the area target.

Mini summary - Translate non price requirements like area or volume into numbers, then compare totals.

Part I - Common errors and quick fixes

• Error 1 - Comparing total price without adjusting for quantity. Fix: always compute unit rate first.
• Error 2 - Forgetting fixed costs in profit problems. Fix: write revenue, variable cost, then add fixed cost before subtracting.
• Error 3 - Using the wrong unit in a unit rate. Fix: state the unit in words inside your calculation (for example, pesos per liter).
• Error 4 - Ignoring rounding up to whole items. Fix: after computing an exact quantity, round up to the next whole bag, bottle, or ticket.
• Error 5 - Forgetting fees in installment totals. Fix: calculate monthly outlay including fee, then multiply by the number of periods.

Checkpoint I1 - Spot and repair the mistake

A learner says 2 kg at ₱130 and 5 kg at ₱320 are the same value because 130 and 320 are both even. What is wrong and how do you fix the comparison

Show Answer

Parity does not matter. Compute pesos per kilogram.

$\text{₱/kg}=\frac{130}{2}=65$

$\text{₱/kg}=\frac{320}{5}=64$

The 5 kg pack is cheaper per kilogram.

Mini summary - A correct solution always shows the numbers that justify the decision.

Part J - Your turn with solve and explain mini sets

J1 - Grocery bundle with delivery fee

A shop can deliver soap bars for a class hygiene drive. Type A is 90 g at ₱28. Type B is 120 g at ₱36. The class needs at least 6 kg total and the delivery fee is ₱200 regardless of quantity. You want the lowest total cost.

Guiding question - compute pesos per gram for each type and decide which to use

Show Answer

Type A unit cost:

$\text{₱/g}=\frac{28}{90}\approx 0.311$

Type B unit cost:

$\text{₱/g}=\frac{36}{120}=0.3$

Type B is cheaper per gram. Use only B to minimize cost.

Guiding question - how many bars of B reach at least 6 kg and what is the total including delivery

Show Answer

Each B is 120 g or 0.12 kg. Bars needed:

$\frac{6}{0.12}=50$

Goods cost:

$50\times 36=1800$

Total with delivery:

$1800+200=2000$

Short explanation: choose B for lower unit cost 0.3 per gram, buy 50 bars to hit 6 kg, and pay ₱2000 total with delivery.

J2 - After school coaching goal

You coach for ₱160 per hour. You can do 3 hours on Wednesday and 2 hours on Saturday each week. Your two month goal is ₱7,500.

Guiding question - compute weekly earnings and then the total for eight weeks

Show Answer

Weekly hours:

$3+2=5$

Weekly earnings:

$160\times 5=800$

Eight weeks total:

$800\times 8=6400$

Guiding question - how many extra hours do you need to reach ₱7,500

Show Answer

Gap is:

$7500-6400=1100$

Hours needed:

$\frac{1100}{160}=6.875$

You need 7 extra hours spread across the two months.

Short explanation: you earn ₱6,400 in eight weeks at your normal schedule and need about 7 extra hours to reach the goal of ₱7,500.

J3 - Photo printing micro business with target profit

You plan to sell 60 photo prints at a school fair. Ink costs ₱260, paper ₱240 for a pack that covers exactly 60 prints, and electricity is ₱80. You want a profit of at least ₱600. What price per print should you set

Guiding question - compute total cost and then solve for price p when profit equals 600

Show Answer

Total cost:

$260+240+80=580$

Profit equation with 60 prints:

$600=p\times 60-580$

Add 580 to both sides:

$1180=p\times 60$

Solve for p:

$p=\frac{1180}{60}=19.666$

Round up to a practical price like ₱20 per print to meet the target.

Short explanation: set price to ₱20 so that revenue 20 × 60 = 1200 minus 580 equals ₱620 profit which meets the target.

J4 - Installment or save then buy with simple interest on savings

A tablet costs ₱12,000 cash. Installment is six months with a fee of ₱60 per month and equal base payments. A bank lets you save ₱2,000 per month with simple interest of 2 percent per year. You can wait six months to buy.

Guiding question - compute the installment monthly outlay and total

Show Answer

Base payment:

$\frac{12000}{6}=2000$

Monthly outlay:

$2000+60=2060$

Total:

$2060\times 6=12360$

Guiding question - estimate interest earned with save then buy and compare totals

Show Answer

A quick Grade 8 estimate uses average balance about half the final principal for half a year. Average principal ≈ ₱6000, time is 0.5 year, rate is 0.02.

$I=6000\times 0.02\times 0.5=60$

Save then buy total payment remains ₱12,000 and you even gain about ₱60 interest. Installment is more expensive by about ₱360 compared with buying cash after saving, not counting the benefit of earlier usage.

Short explanation: if you can wait, save then buy is cheaper by total cost in this case.

J5 - Pass versus pay per ride with partial school days

There are 19 school days this month and 4 club days with extra travel. You ride two times on normal days and four times on club days. Fare is ₱18 per ride. The pass costs ₱720.

Guiding question - compute total rides and decide pass or pay per ride

Show Answer

Normal rides:

$19\times 2=38$

Club day rides:

$4\times 4=16$

Total rides:

$38+16=54$

Pay per ride cost:

$54\times 18=972$

Pass costs ₱720, which saves ₱252 this month.

Short explanation: compute total rides with extra club days, then compare totals. The pass wins at 54 rides.

Part K - Mini reflection and transfer

You have used the same rules in many settings. The path was describe, choose a rule, compute, then explain. As you face new money choices, repeat this path.

• Describe the goal.
• Choose the decision rule.
• Compute the needed values with the right units.
• State the decision and the reason in one to three sentences.

Final Checkpoint - Write and solve one of your own

Pick a real decision in your family this week. It can be a food pack choice, a transport decision, or a school materials plan. Write it in three sentences using the pattern. Include at least one MathML computation. Ask yourself which lens you used and why.

Show Answer

Answers will vary. A complete solution names the goal, shows the unit rate or total cost math, and states a clear decision linked to the numbers.

References

• OECD - Programme for International Student Assessment - Financial literacy frameworks and sample questions.
• Practical Money Skills - Everyday budgeting and unit price activities.
• Basic consumer math texts that cover unit rate, profit and loss, and simple interest.
• School based entrepreneurship modules on break even analysis and pricing.
• Local transport fare schedules for building realistic pass thresholds.

💡 Example in Action

Worked Example 1 - Best Buy with Unit Conversion

Problem. A grocery sells two cooking oil options: Bottle A - 900 ml for ₱132; Bottle B - 1.8 L for ₱250. Which is the cheaper ₱ per liter? If a family needs at least 3.6 liters, how many of the cheaper option should they buy and what total goods cost will they pay (ignore other fees)?

Show Answer

Convert and compute unit prices. A: 0.9 L, ₱/L = 132 ÷ 0.9 = 146.67. B: 1.8 L, ₱/L = 250 ÷ 1.8 = 138.89. Cheaper per liter is B. Need 3.6 L so buy 2 bottles of B. Total goods cost = 2 × 250 = ₱500.

$\text{₱/L}=\frac{132}{0.9}=\mathrm{146.666\dots }$

$\text{₱/L}=\frac{250}{1.8}=\mathrm{138.888\dots }$

$\frac{3.6}{1.8}=2$

$2\times 250=500$

Worked Example 2 - Profit or Loss with Fixed and Variable Costs

Problem. Price ₱25, variable cost ₱11, fixed booth fee ₱420. a) What is the break-even quantity? b) If they sell 90 sets, what is the profit?

Show Answer

Margin = 25 − 11 = 14. BEQ = 420 ÷ 14 = 30. At 90: revenue 90 × 25 = 2250; variable 90 × 11 = 990; total cost 990 + 420 = 1410; profit 2250 − 1410 = 840.

$\text{Margin}=25-11=14$

$\text{BEQ}=\frac{420}{14}=30$

$2250-1410=840$

Worked Example 3 - Installment vs Cash (with Monthly Fee)

Problem. Cash price ₱8,400. Installment plan 6 equal base payments + ₱50 monthly service fee. a) Find base payment, monthly outlay, total paid. b) Find extra vs cash.

Show Answer

Base = 8400 ÷ 6 = 1400. Outlay = 1400 + 50 = 1450. Total = 1450 × 6 = 8700. Extra = 8700 − 8400 = 300.

$\frac{8400}{6}=1400$

Worked Example 4 - Pass Threshold and Decision

Problem. Fare ₱18/ride. Pass ₱720. a) Minimum rides R for pass to be cheaper. b) If 21 days at 2 rides per day, should student buy pass

Show Answer

Threshold: 720 ≤ 18R → R ≥ 40. Rides this month: 21 × 2 = 42. 42 ≥ 40 so buy the pass.

$720\le 18\times R$

Worked Example 5 - Competing Supplier Plans (Setup vs None)

Problem. Plan A: ₱220 each, no setup. Plan B: ₱180 each + ₱800 setup. Compare total cost at q = 40 and q = 70.

Show Answer

q = 40 - A: 220 × 40 = 8800; B: 180 × 40 + 800 = 8000. q = 70 - A: 15400; B: 13400. Plan B cheaper in both.

Now You Try - 5 Tasks

NYT 1 - Best Buy

Cereal: X - 700 g for ₱95; Y - 1.2 kg for ₱156. Which is cheaper per kg

Show Answer

Convert 700 g = 0.7 kg. X: 95 ÷ 0.7 = 135.714… Y: 156 ÷ 1.2 = 130. Cheaper - Y.

NYT 2 - Surplus or Deficit

Income ₱12,500. Planned 4200 + 1800 + 2000 + 1300. Repair 1700. Surplus or deficit

Show Answer

Planned 9300. With repair 11000. Surplus 12500 − 11000 = 1500.

NYT 3 - Profit and Break Even

Price 40, variable 22, fixed 360. a) BEQ b) Profit at 55

Show Answer

Margin 18. BEQ 360 ÷ 18 = 20. At 55: revenue 2200; VC 1210; TC 1570; profit 630.

NYT 4 - Installment

Cash 1260. 3 months. Fee 30 per month. Find base, monthly outlay, total, extra.

Show Answer

Base 420. Outlay 450. Total 1350. Extra 90.

NYT 5 - Pass Threshold

Fare 17, pass 680, month rides: 20 days × 2 + 3 days × 4. Buy pass

Show Answer

Rides 52. Pay per 884 vs pass 680. Buy pass (saves 204).

📝 Try It Out

1) Best Buy with Exact Need

500 g at ₱38 and 1 kg at ₱70. Need 3 kg. Which pack type and minimum goods cost

Show Answer

A (₱/kg) = 38 ÷ 0.5 = 76. B (₱/kg) = 70 ÷ 1 = 70. Choose 1 kg packs. Buy 3 × 1 kg = 3 kg for ₱210.

2) Monthly Surplus or Deficit

Income 3600. Expenses 1480 + 880 + 1050 + 420. Surplus or deficit

Show Answer

Total 3830 → deficit 230.

3) Profit or Loss with Break-even and Profit at a Quantity

Price 35, variable 18, fixed 540. a) BEQ b) Profit at 70

Show Answer

Margin 17. BEQ ≈ 31.76 → 32. Profit at 70 = 650.

4) Installment with Monthly Fee

Cash 4800, 4 months, fee 45. Base, outlay, total, extra

Show Answer

Base 1200. Outlay 1245. Total 4980. Extra 180.

5) Pass Threshold and Month Decision

Fare 19, pass 760. Rides this month 48. Buy pass

Show Answer

Threshold 760 ÷ 19 = 40. 48 ≥ 40, pass saves 152.

6) Simple Interest

Principal 7500, r = 3% per year, t = 8 months. Interest

Show Answer

t = 8/12 = 2/3. I = 7500 × 0.03 × 2/3 = 150.

7) Supplier Choice with Delivery (Find the Crossover)

Plan A 24 each free delivery. Plan B 20 each + 320 delivery. Compare at q = 40 and q = 80.

Show Answer

q = 40 → A 960, B 1120 - A cheaper. q = 80 → A 1920, B 1920 - tie. Beyond 80 B wins.

8) Target Profit - Solve for Price

q = 60, variable 26, fixed 680, target profit 1000. Price

Show Answer

VC 1560. TC 2240. 1000 = p × 60 − 2240 → p = 54.

9) Data Plan Combination (Exact Need vs Unit Rate)

Plan A 10 GB for 280. Plan B 6 GB for 180. Need 18 GB. Cheapest combination

Show Answer

Two As = 20 GB, ₱560. Three Bs = 18 GB, ₱540. Choose 3 × B.

10) Paint Mix - Combination for Minimum Cost

Need 7 L. 4 L at 420 and 1 L at 120. Cheapest combo and total cost

Show Answer

1 × 4 L + 3 × 1 L = 7 L at 420 + 360 = 780 (cheapest).

✅ Check Yourself

1) Best Buy (unit conversion) - 750 ml for ₱115 vs 1 L for ₱150. Which is cheaper per liter

Show Answer

0.75 L: 115 ÷ 0.75 = 153.33; 1 L: 150 ÷ 1 = 150. Cheaper - 1 L at ₱150/L.

2) Threshold decision (movies) - ₱80 per movie vs pass ₱600. a) Minimum m for pass b) If 9 movies, what to choose

Show Answer

600 ≤ 80m → m ≥ 7.5 → 8. At 9, pass saves: 720 vs 600 - buy pass.

3) Surplus or deficit with a shock - Income 10500; planned 4300 + 1600 + 3800 + 900; repair 700. Surplus or deficit

Show Answer

Planned 10600. With repair 11300. Deficit 800.

4) Profit and break-even - Price 55, variable 31, fixed 960. a) BEQ b) Profit at 70

Show Answer

Margin 24. BEQ 40. Profit at 70: R 3850; VC 2170; TC 3130; P 720.

5) Installment vs cash - Cash 6000; 5 months; fee 65. Base, outlay, total, extra

Show Answer

Base 1200; outlay 1265; total 6325; extra 325.

6) Pass threshold with mixed days - Fare 17; pass 700; rides 20 × 2 plus 3 × 4. Buy pass

Show Answer

Threshold 700/17 ≈ 41.18 → 42 rides. Usage 52 rides, pay per 884. Buy pass 700.

7) Target profit to price - q 90; variable 28; fixed 1040; profit 1500. Minimum price

Show Answer

VC 2520; TC 3560; 1500 = p × 90 − 3560 → p = 56.222… → set at least 57.

8) Supplier choice with setup and minimum order - A 210 each; B 170 each + 1600 setup; min 60. a) Crossover b) With min, when is B cheaper

Show Answer

170q + 1600 ≤ 210q → q ≥ 40. With min 60, B usable only if q ≥ 60 and is cheaper for all such q.

9) Simple interest - P 9000, r 2.5% per year, t 15 months. I

Show Answer

t = 15/12 = 1.25. I = 9000 × 0.025 × 1.25 = 281.25.

10) Best buy with wastage - Need exactly 3 kg. 2 kg at 120 or 5 kg at 280. Leftover spoils. Which to buy

Show Answer

Two 2 kg = 4 kg for 240 (use 3, waste 1) → 80 per used kg. One 5 kg = 5 kg for 280 (use 3, waste 2) → 93.33 per used kg. Choose two 2 kg.

11) Fuel and rental decision - Trip 420 km. A 12 km/L. B 16 km/L plus ₱800 higher rental. Fuel ₱70/L. Which is cheaper

Show Answer

A liters 35 → ₱2450. B liters 26.25 → ₱1837.5. Fuel saving 612.5. Rental +800. B is 187.5 more expensive. Choose A.

12) Budget adjust to hit a savings goal - Income 3800; current total 3900; want save 200. How much reduce and one valid adjustment

Show Answer

Needed expenses 3600. Reduce by 300. Example - cut leisure 200 and food 100.

13) Bulk discount effect - Price 30; variable 16 or 14 if ≥ 80; fixed 600; plan 80 units. Extra profit from bulk

Show Answer

No bulk profit 520; with bulk 680; extra 160.

14) Data plan combination (integer choice) - A 12 GB 300; B 5 GB 160; need ≥ 27 GB. Cheapest combo

Show Answer

1A+3B = 27 GB for 780; 2A+1B = 29 GB for 760 - cheaper. Choose 2A+1B.

15) Concept classification - Match each to a lens: a) pass minimum rides b) compare price per kg c) will income cover expenses d) how many units to cover fee e) compare 4 month installment vs cash

Show Answer

a) Threshold b) Best buy c) Surplus or deficit d) Break even e) Installment vs cash.

🚀 Go Further

1) Two Siblings, One Month, How Many Passes

Sibling A: 21 school days × 2 rides, 3 club days × 2 extra. Sibling B: 18 school days × 2, 6 club days × 2 extra. Fare ₱18/ride. Pass ₱720 per person (up to 60 rides). Choose 0, 1, or 2 passes to minimize total family cost.

Show Answer

A rides 48; B rides 48; threshold 40 rides each. Costs - 0 pass: 96 × 18 = 1728; 1 pass: 720 + 48 × 18 = 1584; 2 passes: 1440. Minimum - 2 passes.

2) Price with Buy 5, Get 1 Free

Variable 18 each, fixed 540. Promo makes 1 of 6 free. Forecast 120 distributed. Target profit 1200. Find price.

Show Answer

Paid units 100. VC for 120 is 2160; TC 2700. 1200 = 100p − 2700 → p = 39.

3) Installment with Tiered Fees vs Save-Then-Buy

Cash 12600. 6 months. Fee 90 for months 1–3, fee 45 for months 4–6. Compare total with save-then-buy (no interest).

Show Answer

Base 2100. Total installment 2190 × 3 + 2145 × 3 = 13005. Save-then-buy 12600. Installment is 405 more for earlier access.

4) Bulk vs Retail with Possible Spoilage

Retail 5 kg for 270; bulk 25 kg for 1250 with 10% spoilage risk over two weeks. Household uses 12 kg/week. Compare effective ₱ per used kg over two weeks.

Show Answer

Retail need 24 kg → cost 24 × 54 = 1296. Bulk usable 22.5 kg → effective 1250/22.5 = 55.56 per kg, worse than retail 54. Choose retail unless spoilage is prevented.

5) Shared Fixed Cost with Two Products

Shared fixed 900. Item A margin 16, item B margin 25. Profit P = 16a + 25b − 900. Find one combination with P ≥ 700.

Show Answer

Need 16a + 25b ≥ 1600. Try b = 20 → need 1100 from A → a ≥ 68.75 → 69. Contribution 1104 + 500 = 1604 → P = 704 ≥ 700. One feasible solution is (a = 69, b = 20).

🔗 My Reflection

Write your responses in your notebook. Choose the 3-2-1 format below. Keep each bullet to 1-3 sentences and include one small MathML computation from a problem you solved today.

3 - Key ideas you are taking with you

2 - Strategies you will use next week

1 - Question you still have (or a scenario you want to try next)

-

Sentence frames you can use

• I can decide between two options by computing the unit rate $\text{₱/unit}=\frac{\mathrm{Price}}{\mathrm{Quantity}}$ and choosing the lower value.
• To check if a pass is worth it, I solve $\text{Pass}\le \text{Rate}\times R$ for $R$.
• For profit, I compute $P=\mathrm{Revenue}-\mathrm{Cost}$ after adding fixed and variable costs.

Include one tiny MathML snippet from your work

• Unit rate you computed today: $\text{₱/kg}=\frac{250}{1.8}\approx 138.89$
• Threshold you solved: $R\ge \frac{720}{18}=40$
• Profit you checked: $P=2250-1410=840$

Show Answer (sample reflection)

3 I learned (1) to compare options fairly using unit rate $\text{₱/L}=\frac{250}{1.8}\approx 138.89$, (2) to find a pass threshold with an inequality, and (3) to include fixed costs before deciding profit.

2 Next week I will (1) write the rule first (unit rate, threshold, or profit), then compute; (2) check limits like minimum order or spoilage before I decide.

1 I still wonder: if two plans tie in total cost, which tie-breakers should I use (quality, risk, or time)?