Turning a described situation into an equation of that shape

0:00Here is a floor that has to be carpeted, and everything you are told about it. The carpet is to cover three hundred square metres, and the hall is to be one metre longer than twice its breadth. You are not asked to solve anything yet. You are asked to turn that into an equation. And a modelling question is always the same three moves, in the same order. Choose one unknown. Write every other quantity in the story in that letter. Then find the one sentence that equates two things you can now both write down.

0:38Start with the choice of letter, because the commonest mistake happens before any algebra does. The floor has a breadth and a length. Two quantities, so it feels natural to take two letters. It is not natural. It is a loss. The story says the length is one more than twice the breadth. That sentence is information, and calling the length y throws it away. Let the breadth be x metres. Then the length is not a second unknown at all: it is two x plus one.

1:13One letter is enough exactly when a sentence ties the quantities together — a stated rule, or a stated total — and every situation here comes with one. So: breadth x, length two x plus one. Notice that nothing has been solved and nothing has been assumed. The second expression was read straight off the sentence. Now the area. Area is breadth times length, which is x times, bracket, two x plus one.

1:48Multiply it out: two x squared plus x. There is the square, and it arrived without being asked for. Now the third move, and it is worth being slow about. Look for the sentence that states a FACT about the situation, rather than the one telling you what to report. 'Find the length and the breadth' is not that sentence. That is the question, and questions do not become equations.

2:18'The carpet is to cover three hundred square metres' is that sentence. It states a quantity you have just learned how to write. So two x squared plus x equals three hundred. Move the three hundred across and you have it: two x squared plus x minus three hundred equals nought. A twelve metre breadth gives a twenty-five metre length, and twelve times twenty-five is three hundred exactly. Here is the part the working never says out loud.

2:52The square did not appear because the problem is hard. It appeared because two quantities that BOTH move with x were multiplied together. The breadth moves with x. The length moves with x. Multiply them and one x meets another, and an x squared is what that is. You can watch it happen without any story at all. Take every product of two linear expressions — three thousand nine hundred and sixty-nine of them — and file each by how many of its factors actually move.

3:28When both factors move, the product is of degree two. Every single time, all two thousand nine hundred and sixteen of them. When one factor is fixed, it never happens. Not once. Now the contrast, on the same floor: ask for the PERIMETER instead of the area. Perimeter is two times, bracket, breadth plus length. The breadth moves, the length moves, but the TWO does not. Two times, bracket, x plus two x plus one, is six x plus two. Linear. No square anywhere.

4:05Same floor, same letter. Different sentence, and the square is gone. Second situation, and this one hands you a total rather than a rule. Two children have forty-five marbles between them. Each of them loses five. The two counts that are left multiply to a hundred and twenty-four. Let the first child's count be x. Then the second is forty-five minus x — and that step deserves the question people skip: why is that allowed?

4:39Because the two counts have to add to forty-five. Name one, and the other is what is left. There is nothing to choose. After the loss the counts are x minus five, and forty minus x — forty-five minus x, less another five. Their product expands to minus x squared plus forty-five x minus two hundred. Set that equal to a hundred and twenty-four and gather. Multiply through by minus one, and there it is.

5:13Two moving factors again: one child's remainder times the other's, both written in x. There are forty-six ways to split forty-five marbles, and exactly two leave a product of a hundred and twenty-four. Nine and thirty-six. Nine leaves four and thirty-one, whose product is a hundred and twenty-four; thirty-six is the same pair the other way about. Third situation, and the moving factor here is a price. A workshop makes some number of toys in a day, and the cost of making one, in euros, is fifty-five reduced by that day's output.

5:56That is the sentence doing the work. Make more, and each one costs less. Let x be the number made. Then the cost of one is fifty-five minus x, and the day's total bill is x times, bracket, fifty-five minus x. A count times a price, and the price is written in the count. Both factors move. The total that day was seven hundred and fifty euros. Expand and flip the sign: x squared minus fifty-five x plus seven hundred and fifty equals nought.

6:33Of the fifty-six possible outputs, exactly two cost seven hundred and fifty: twenty-five toys at thirty euros each, or thirty toys at twenty-five euros each. Both of those needed multiplying through by minus one, and it is worth stopping on that for a moment. Of the seven equations here, five come out with a positive x squared on their own; two do not, and both get flipped. The flip is cosmetic. It is done so the standard form reads with a positive leading coefficient, and for no other reason.

7:14Does it move any answer? No — and because that is a statement about two SETS of numbers, it was walked rather than asserted: every value on the grid asked twice, of the original and of the flipped one. They agreed everywhere. Nothing was lost and nothing was invented. Multiplying by nought is a different story: it invents solutions by the thousand, because nought equals nought is true wherever you stand.

7:45Which is why that is the one multiplier you may not use. Four more to build, and the first two are quick. A rectangular plot has an area of five hundred and twenty-eight square metres, and its length exceeds double its breadth by one metre. That is the hall again with a different number. Breadth x, length two x plus one, and two x squared plus x minus five hundred and twenty-eight equals nought.

8:17Sixteen metres by thirty-three. Next: two consecutive positive integers whose product is three hundred and six. Consecutive means the second is one more than the first — that is the tying sentence. Call the smaller x and the larger is x plus one. x times, bracket, x plus one, equals three hundred and six, so x squared plus x minus three hundred and six equals nought. Seventeen and eighteen. The third is an age problem, and it contains the commonest slip in the topic.

8:58A child is twenty-six years younger than their mother, and in three years the two ages will multiply to three hundred and sixty. Let the child's present age be x. The mother is x plus twenty-six. Now go forward three years. The child's age becomes x plus three. And here is the slip: the mother's age does NOT stay at x plus twenty-six. Three years pass for the mother as well.

9:27The mother's future age is x plus twenty-nine. So the product is x plus three, times x plus twenty-nine, which expands to x squared plus thirty-two x plus eighty-seven. Set it equal to three hundred and sixty and gather. The child is seven and the mother thirty-three. In three years, ten and thirty-six, and ten times thirty-six is three hundred and sixty. Had the mother been left at x plus twenty-six, nothing would have fitted.

10:03The fourth is different in kind — the exception to everything said so far. A train covers four hundred and eighty kilometres at a steady speed. Had the speed been eight kilometres an hour less, the journey would have taken three hours longer. Let the speed be x. Time is distance over speed, so the journey takes four hundred and eighty over x hours. The slower journey takes four hundred and eighty over x minus eight.

10:35Which of the two is bigger? The slower one — a slower train takes longer over the same distance, and getting that the right way round is what fixes where the three goes. So the slower time minus the actual time is three. There is no product of two moving quantities anywhere in that sentence. The unknown is in a denominator instead. Multiply through by x and by x minus eight to clear the fractions, and the square arrives from the clearing itself.

11:09Divide by three and gather. Forty kilometres an hour. That journey takes twelve hours; at thirty-two it takes fifteen; and fifteen is three more than twelve. One last thing, and it is why forming the equation is not the same as answering the question. An equation is a statement about numbers. It knows nothing about the story it came from. The hall's equation is perfectly happy with a breadth of minus twelve and a half metres, and the train's with a speed of minus thirty-two kilometres an hour. Neither of those exists.

11:51Across the seven situations, these equations allow fourteen numbers in all, and five of them are numbers no situation could use. Every one is thrown out by a condition the situation carries and the equation does not record: a length is positive, a count is whole, an age is positive, and a speed must be more than eight or the slower train is going backwards. So write the conditions beside the equation, while you still remember where they came from.

12:27A word on the checking, because the output of this topic is an equation — and an equation can simply be copied. Getting the same equation back proves nothing about whether it describes the story. So every situation was asked in two directions that share no machinery. The STORY was asked at a value, in ordinary arithmetic with no algebra at all: with nine marbles, are there really a hundred and twenty-four left?

13:00The EQUATION was asked at the same value: does the standard form come out at nought here? The two answers were filed together as one key, over fifteen hundred and sixty-five values, and both disagreeing cells came out empty. Which on its own proves nothing, because a comparison that had stopped comparing would look exactly the same. So beside each honest model ran a MIS-MODELLED one, through the same code with only a tag changed: only one child loses five marbles, only the child ages, the faster train takes longer.

13:42Those tables are not empty. Sixteen values separate the mis-modelled stories from their equations, and that occupied cell is what makes the empty ones evidence. The coefficients were reached four ways as well: by multiplying out, by the form we wrote down, by a route that never multiplies anything but evaluates the situation at three values and solves for the curve through them, and by a fourth that forgets the cross terms and has to disagree.

14:18Choose one unknown, and only one, because a sentence in the story ties the quantities together. Write every other quantity in that letter — including the ones you get by subtracting from a stated total. Find the sentence that states a fact, not the sentence that asks the question, and make that the equals sign. The square appears when two quantities that both move with the unknown are multiplied. Name the multiplication and it stops being a surprise.

14:53It can also arrive from clearing a denominator, which is the other route worth knowing. Tidy into standard form, flipping the sign if you need to, because that moves no answer. And write down what the equation cannot: that lengths are positive, counts are whole, and speeds are real.