Year 8NumberFinancial mathematicsYear 8 Maths

Comparing money plans, and taking a discount off

What this lesson covers

  • Solutions to problems involving New Zealand currency are rounded to two decimal places.
  • Cash payments in New Zealand are rounded up or down to the nearest 10 cents.
Key idea

Which of these is cheaper?

  • One plan costs $12 a week.
  • Another costs $50 a month.
  • You genuinely cannot tell. Not yet.
PlanChargedCost
Plan Aweekly$12
Plan Bmonthly$50
Cheaper?not yet answerable
Key idea

Because they are on different clocks

  • $12 and $50 are not measuring the same thing.
  • One is a week of cost, the other is a month of it.
  • Picking the smaller number is picking the shorter period.
PlanChargedCost
Plan Aweekly$12
Plan Bmonthly$50
Cheaper?not yet answerable
Worked example

So put both on the same clock

  • A year works for everything. $12 × 52 is $624.
  • $50 × 12 is $600.
  • Now they are comparable, and the monthly plan is cheaper.
PlanWorkingPer year
$12 a week× 52$624 a year
$50 a month× 12$600 a year
So Plan Bis cheaper by$24 a year
Key idea

The shortcut that costs you $48

  • It is tempting to say $12 a week is $48 a month.
  • That gives $576 a year. But the real figure is $624.
  • $48 has gone missing, and nothing in the working shows it.
MethodWorkingGives
$12 × 4assumes 4 weeks a month$48 a month
That is× 12$576 a year
The truth52 weeks, not 48$624 a year
Missing$48
Key idea

A month is not four weeks

  • There are 52 weeks in a year and 12 months.
  • So a month is 52 ÷ 12 weeks, which is about 4.33.
  • Four weeks a month would only fill 48 of the 52.
FactWorkingValue
Weeks in a year52
Months in a year12
So one month is52 ÷ 124.33 weeks
Key idea

Go via the year and never touch 4.33

  • Weekly to yearly: multiply by 52.
  • Yearly to monthly: divide by 12.
  • So $12 a week is really $52 a month. Both steps are exact.
StepWorkingGives
Weekly to yearly× 52$624
Yearly to monthly÷ 12$52
So $12 a weekreally is$52 a month
Worked example

The same move, saving instead of spending

  • You put away $25 a week for 20 weeks.
  • 25 × 20 is $500.
  • A savings plan is a spending plan with the sign flipped.
PlanWorkingTotal
$25 a week, 20 weeks25 × 20$500
$25 a week, a year25 × 52$1300
$100 a month, a year100 × 12$1200
Worked example

When a plan has two parts

  • Plan A: $30 a month, plus 20c a minute past 100 minutes.
  • Plan B: $45 a month, flat.
  • Which is better? It depends how much you talk.
UsageWorkingCost
150 minutesA: 30 + 50 × 0.20$40
B: flat$45
Worked example

And they swap places

  • At 150 minutes, A costs $40 and B costs $45. A wins.
  • At 200 minutes, A costs $50 and B costs $45. B wins.
  • So "which plan is cheaper" has no answer on its own.
UsageWorkingCost
150 minutesA: 30 + 50 × 0.20$40
B: flat$45
200 minutesA: 30 + 100 × 0.20$50
B: flat$45
Key idea

Where exactly do they cross?

  • A costs $15 more than its base before it catches B.
  • $15 at 20c a minute is 75 minutes of calls.
  • So they are level at 175 minutes. Past that, take B.
StepWorkingGives
Gap to close45 − 30$15
At 20c a minute15 ÷ 0.2075 minutes
Past the first 100100 + 75175 minutes
Key idea

Buy now, pay later: the plain arithmetic

  • A $180 jacket, four payments of $45.
  • 4 × 45 is 180. Pay all four on time and it cost $180.
  • That is genuinely all it cost. No interest was added.
SituationWorkingTotal
A $180 jacket4 payments of $45$180
If all four are paidtotal cost$180
Worked example

Now miss one

  • Say a missed payment carries a $10 fee.
  • The jacket now costs $190 instead of $180.
  • Which is 5.6% more, for being late once.
SituationWorkingTotal
A $180 jacket4 payments of $45$180
If all four are paidtotal cost$180
One missed, $10 feetotal cost$190
Worked example

Discounts, on bigger numbers now

  • A $180 coat with 35% off.
  • 35% of 180 is 0.35 × 180, which is $63.
  • So you pay $180 − $63, which is $117.
StepWorkingGives
35% of $1800.35 × 180$63
$180 − $63$117
Key idea

Or skip the subtraction

  • 35% comes off, so you pay the other 65%.
  • 0.65 × 180 is $117, straight away.
  • Same answer, one calculation, one fewer chance to slip.
StepWorkingGives
35% of $1800.35 × 180$63
$180 − $63$117
Or straight to it65% of $180$117
Key idea

And the two money rules, still true

  • Money is always written with two decimal places.
  • A cash total rounds to the nearest 10 cents.
  • $56.86 becomes $56.90 in cash, and stays $56.86 on a card.
PaymentRuleYou pay
$56.86 by cardexact cents$56.86
$56.86 in cashnearest 10c$56.90
Always writtentwo decimal places
Your turn

Your turn: four questions

  • Pause here and try all four.
  • Question 2 is the one with 52 in it, not 48.
  • Question 4 is simpler than it looks.
Question
1A $180 coat has 35% off. What do you pay?
2You save $15 a week. How much in a year?
3A cash total is $56.86. What do you hand over?
4Four payments of $45. What is the total cost?
Show answer

1. $117. 35% of $180 is $63, and $180 − $63 = $117. Or 65% of $180 in one step. 2. $780. That is 52 × $15, not 48 × $15. 3. $56.90. The 6 cents rounds up to the nearest 10 cents. On a card it would stay $56.86. 4. $180, provided all four are paid on time. The plan itself adds nothing; the cost appears only if a payment is missed.

Worked example

The answers

  • 35% off $180 leaves $117, or 65% of $180 in one step.
  • $15 a week is $780 a year. And $56.86 cash is $56.90.
  • Four payments of $45 is $180, if they are all on time.
Answer
1$117
2$780
3$56.90
4$180
Summary

What to take away

  • Two plans on different clocks cannot be compared. Convert first.
  • Go weekly to yearly to monthly. A month is not four weeks.
  • And 35% off means paying 65%, in a single step.
Remember
Different periodscannot be compared as they stand
Use the yearas the common ground
A monthis 4.33 weeks, never 4
35% offmeans pay 65%, in one step
Cashrounds to 10c; card does not

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