Mathematics Year 7: Solving a problem: from the table to the equation
Mathematics, Year 7 (Portugal): The same three stages, now with an unknown you can name with a letter: compare a table with an equation, and always check the answer against the situation.
A situation with an unknown
A swimming club has two plans. Plan A: 15 € once, plus 4 € per visit. Plan B: 7 € per visit, nothing else. You know the stages: interpret (what is known? what is asked?), choose and carry out a strategy, evaluate. Now the unknown gets a letter: v is the number of visits, so A costs 15 + 4 × v and B costs 7 × v. From this situation you can pose problems yourself: when do both cost the same? With 60 €, how many visits does each plan give?
Two strategies for “when do they cost the same?”
Strategy 1, a table (a spreadsheet does it for you): for v = 1 to 6, plan A gives 19, 23, 27, 31, 35, 39 and plan B gives 7, 14, 21, 28, 35, 42. At v = 5 both cost 35 €, and the table also shows who wins before and after. Strategy 2, an equation: 15 + 4v = 7v, so 15 = 3v and v = 5. Both are correct, and different: the table shows, the equation shortens. With a 150 € fee the table would need 50 rows, while 150 = 3v gives v = 50 at once. Here the equation is more efficient.
A case: 60 € to spend
Which plan gives more visits for 60 €? Interpret: v must be a whole number of visits. Plan A: 15 + 4v = 60, so 4v = 45 and v = 11.25. Plan B: 7v = 60, so v = 60 ÷ 7 = 8.57… Evaluate: nobody makes 0.25 of a visit, so A allows 11 visits (15 + 44 = 59 €) and B allows 8 (7 × 8 = 56 €). Check the next one: the 12th visit in A costs 63 € and the 9th in B also 63 €, both over 60 €. Plan A gives 3 more visits.
Answering the equation, not the problem
Rui solved 7v = 60, got 8.57… and answered “about 9 visits”, rounding to the nearest whole number. It looks reasonable, but 9 visits cost 9 × 7 = 63 € and he only has 60 €. The number that comes out of an equation answers the equation; the evaluate stage answers the problem. Here visits are whole and the money is limited, so you round down: 8 visits. To avoid the trap, always ask: does this value make sense in the situation, and does it respect what the problem says?
Where you see it: a fixed cost plus a cost per unit
Hiring a hall for 90 € plus 6 € per guest, renting a bike with a starting fee, a phone plan with a monthly fee: all have this shape, a fixed part plus a part that grows. With 150 € for the hall: 90 + 6g = 150, so 6g = 60 and g = 10 guests. In a spreadsheet you type =90+6*A2 and drag it down, then compare the plans row by row. The technology does the arithmetic; you pose the question and evaluate whether the answer fits.
Test what you learned
Ana solved “when do the two plans cost the same?” with a table, Rui with an equation. Both found 5 visits. Which sentence is true?
- a) Only the equation is right; a table is just guessing
- b) Both are correct and different: the table shows more, the equation is shorter with big numbers
- c) They are the same strategy, because both give the same answer
- d) Only the table is right, because equations use letters and letters are not numbers
Correct answer: b) Both are correct and different: the table shows more, the equation is shorter with big numbers — Right: same correct result, two different routes. With a 150 € fee the equation is far quicker than 50 table rows.
A bike rental charges 10 € once plus 5 € per hour. Ana has 60 €. For how many hours can she rent the bike?
- a) 12
- b) 14
- c) 10
- d) 50
Correct answer: c) 10 — Right: 60 − 10 = 50 € is left for hours, and 50 ÷ 5 = 10. Check: 10 + 5 × 10 = 60.
Solving 7v = 60 for the number of visits gives v = 8.57… Visits are whole and Rui has only 60 €. What is the answer to the problem?
- a) 8 visits
- b) 9 visits, rounding 8.57 to the nearest whole number
- c) 8.57 visits, exactly what the equation gives
- d) 9 visits, using the 4 € left after 8 visits to complete a ninth
Correct answer: a) 8 visits — Right: 8 × 7 = 56 € fits in 60 €, while a ninth visit would bring the total to 63 €.
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