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Comparing two patterns

Patterns and sequences · level 4, lesson 5 of 6

A question to start

Your family offers you a choice of two savings plans.

  • Plan A: you get 50 dollars today, and then 5 dollars more every week.
  • Plan B: you get nothing today, but 10 dollars every week.

Which plan gives you more after 4 weeks? After 10 weeks? After a year? Most people grab Plan A because of the big start. Work it out, and you find something surprising at week 10, and after that one plan wins forever.

The idea

Two sequences, side by side

Each plan is a sequence: the money you have after \(w\) weeks. (We use \(w\) for weeks instead of \(n\), but it plays the same part as the position. Week 0 is today.)

  • Plan A: start 50, add 5 each week. Rule: \(A_w = 50 + 5w\).
  • Plan B: start 0, add 10 each week. Rule: \(B_w = 10w\).

Put them in one table, and add a column for the gap: how far A is ahead of B.

Week \(w\) 0 2 4 6 8 10 12 14
Plan A 50 60 70 80 90 100 110 120
Plan B 0 20 40 60 80 100 120 140
A − B (gap) 50 40 30 20 10 0 −10 −20

After 4 weeks, A is ahead, 70 to 40. After 10 weeks, they are exactly equal: 100 dollars each. After that, B is ahead, and the gap keeps growing in B's favour. After a year (52 weeks) A has \(50 + 260 = 310\) dollars and B has 520.

Why the gap closes steadily

Every week, A gains 5 and B gains 10. So B catches up by \(10 - 5 = 5\) dollars a week. The gap starts at 50, and shrinks by 5 each week: 50, 45, 40, ... The gap is itself a sequence with a constant difference (\(-5\)), and it reaches 0 after

\[50 \div 5 = 10 \text{ weeks.}\]

This is the key idea of the lesson: to compare two straight-line patterns, look at the gap; it changes by the difference of the two differences.

Finding the meeting point with an equation

The same answer comes from asking "when are they equal?":

\[50 + 5w = 10w\]

Take \(5w\) from both sides: \(50 = 5w\), so \(w = 10\). Both methods are really the same thinking: the equation says "the head start (50) equals the catching-up (5 a week) times the weeks".

The picture: two lines crossing

0 2 4 6 8 10 12 14 0 25 50 75 100 125 150 weeks dollars saved week 10: both 100 Plan A: 50 + 5 a week Plan B: 10 a week

On a graph, each plan is a straight line (lesson 3). Plan A starts higher (at 50) but climbs less steeply (5 per week). Plan B starts at 0 but is steeper (10 per week). The steeper line must eventually catch the other, and they cross at the point \((10, 100)\): week 10, 100 dollars. To the left of the crossing A is higher; to the right B is higher, and stays higher, because a steeper line never falls back below a less steep one once it has passed it.

So the full answer to "which is better?" is: it depends how long you wait. Under 10 weeks, Plan A. Exactly 10, a tie. Over 10, Plan B.

When they never meet, and when they meet between whole numbers

  • Same difference, different start (like \(3n + 2\) and \(3n + 7\)): the gap never changes, so they never meet. The lines are parallel.
  • A crossing that is not a whole number. Compare \(25 + 3w\) and \(7w\). The gap is 25 and closes by 4 a week: \(25 \div 4 = 6.25\) weeks. There is no week 6.25 in the table. At week 6 the first is ahead (43 to 42); at week 7 the second is ahead (46 to 49). So the answer is "the second plan is ahead from week 7 on".

Worked examples

1. Phone plans. Plan X costs 20 dollars a month plus 2 dollars per gigabyte (GB) of data. Plan Y has no monthly fee but costs 8 dollars per GB. For how many GB is each plan cheaper?

  • Costs: X is \(20 + 2g\), Y is \(8g\), where \(g\) is the number of GB.
  • Y starts 20 cheaper, but costs 6 more per GB. Equal when \(20 = 6g\), so \(g = 3.33...\)
  • Check whole numbers: 3 GB: X costs 26, Y costs 24 (Y cheaper). 4 GB: X costs 28, Y costs 32 (X cheaper).
  • So: 3 GB or less, Plan Y; 4 GB or more, Plan X.

2. Two candles. Candle A is 30 cm tall and burns down 2 cm an hour. Candle B is 20 cm tall and burns 1 cm an hour. When are they the same height?

  • After \(h\) hours: A is \(30 - 2h\), B is \(20 - h\). Both are sequences going down.
  • A starts 10 cm taller but loses 1 cm more per hour, so the gap of 10 closes in \(10 \div 1 = 10\) hours.
  • After 10 hours: A is \(30 - 20 = 10\) cm, B is \(20 - 10 = 10\) cm. ✓ After that, A is shorter.

3. Living on Mars time. A day on Mars, called a sol, is about 24 hours and 40 minutes long. When a rover lands, its team on Earth often works on Mars time for the first months, so each shift starts about 40 minutes later (by Earth clocks) than the one before. A shift starts at 8:00 in the morning on sol 1. On which sol does it first start at 8:00 in the evening or later?

  • Start time in minutes after 8:00 a.m. on sol \(s\): \(40(s - 1)\). (Sol 1 has had no delays yet.)
  • 8:00 p.m. is 12 hours, 720 minutes, after 8:00 a.m. So we need \(40(s - 1) \ge 720\), which means \(s - 1 \ge 18\).
  • Sol 19 starts at exactly 8:00 p.m. by Earth clocks. (Teams really did live like this, and found it very tiring.)

4. Two number sequences. When does 5, 9, 13, ... (\(4n + 1\)) first pass 50, 52, 54, ... (\(2n + 48\))?

  • \(4n + 1 = 2n + 48\) gives \(2n = 47\), \(n = 23.5\). Not a whole number.
  • At \(n = 23\): \(93\) and \(94\), the second still ahead. At \(n = 24\): \(97\) and \(96\). The first sequence passes the second at position 24.

Try it

  1. Plan C gives 30 dollars today and 4 dollars a week; Plan D gives 7 dollars a week from nothing. When are they equal, and how much is each worth then?
  2. In question 1, which plan is better after 5 weeks? After 20 weeks?
  3. Candle P is 24 cm and burns 3 cm an hour. Candle Q is 18 cm and burns 1.5 cm an hour. When are they the same height, and how tall?
  4. Do the sequences \(3n + 7\) and \(3n + 2\) ever have the same term at the same position? Explain.
  5. A climbing gym charges 15 dollars a visit, or a 60 dollar membership plus 5 dollars a visit. How many visits make the membership worth it?
  6. Two sequences: 100, 97, 94, ... and 10, 13, 16, ... At which position are they equal? What is the term there?
  7. On Mars time, a shift starts at 8:00 a.m. on sol 1, and 40 minutes later each sol. At what time does it start on sol 10?
  8. Plan E gives 100 dollars today and 2 dollars a week. From which week on is Plan B (10 dollars a week from nothing) ahead of Plan E?
Answers
  1. The gap of 30 closes by \(7 - 4 = 3\) a week: \(30 \div 3 = 10\) weeks. Each is worth \(7 \times 10 = 70\) dollars (and \(30 + 40 = 70\) ✓).
  2. Week 5: C has \(30 + 20 = 50\), D has 35, so C. Week 20: C has \(30 + 80 = 110\), D has 140, so D.
  3. \(24 - 3h = 18 - 1.5h\). P starts 6 cm taller and loses 1.5 cm an hour more, so \(6 \div 1.5 = 4\) hours. Then P is \(24 - 12 = 12\) cm and Q is \(18 - 6 = 12\) cm.
  4. No. At every position the first is exactly 5 more than the second; the gap never changes, because both go up by 3. Their graphs are parallel lines.
  5. Pay-per-visit costs \(15v\); membership costs \(60 + 5v\). Equal when \(10v = 60\), \(v = 6\). At 6 visits they both cost 90 dollars. For 7 or more visits the membership is cheaper (7 visits: 105 against 95).
  6. Rules: \(103 - 3n\) and \(3n + 7\). Equal when \(103 - 3n = 3n + 7\), so \(96 = 6n\) and \(n = 16\). The term is \(3 \times 16 + 7 = 55\) (and \(103 - 48 = 55\) ✓).
  7. Sol 10 has had 9 delays: \(9 \times 40 = 360\) minutes, which is 6 hours. It starts at 2:00 p.m.
  8. The gap of 100 closes by \(10 - 2 = 8\) a week: \(100 \div 8 = 12.5\). At week 12, E has 124 and B has 120; at week 13, E has 126 and B has 130. B is ahead from week 13 on.

Common mistake

Comparing at only one moment. "Plan A is better" is true at week 4 and false at week 20. When two patterns grow at different rates, the honest answer names the crossing point and says which is better on each side of it.

Forgetting that positions are whole numbers. If the equation gives \(n = 23.5\), there is no term there. Check the whole numbers on either side to see where one sequence first passes the other.

Where it leads