PrepShorts · Study sheet · Class 7 Mathematics · Chapter 8, Working with Fractions
Chapter 8 · Working with Fractions
Why multiplying can make a number smaller
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“Does multiplying make a number bigger?” is not a question with an answer. It is a question missing a condition.
The idea
Multiplication scales; it does not enlarge. Whether a product comes out above or below the number you started with is decided by 1, not by the operation: a factor above 1 stretches, a factor below 1 shrinks, and 1 itself leaves things alone. So "does multiplying make it bigger?" is not a question with an answer — it is a question missing a condition, and the chapter refuses to supply the answer directly. It builds three cases, leaves the conclusion as blanks for the student to fill, and phrases even that conclusion one factor at a time. The one-factor-at-a-time form is the careful part: a product is below both factors only when both of them sit below 1.
What you should be able to do
- Predict, before computing, whether a product will land above or below each of its factors, from where those factors sit relative to 1
- Produce an example of each of the three printed situations and check the prediction against the computed product
- Compare a product with a factor by rewriting both over a common denominator
- State the chapter's conclusion in its own careful form — a comparison with the other factor, one factor at a time
- Say exactly when a product falls below both of the numbers multiplied, and give a counter-example to the claim that it always does
- Explain why interchanging the two factors cannot change the product, using the rectangle picture
- Recognise that 1 is the hinge of the whole discussion, and why the chapter sets it aside before starting
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| product | the number a multiplication produces | printed throughout §8.1, pp.184–186 |
| counting numbers | the whole numbers used for counting, from 1 upward | printed in §8.1, p.184 |
| between 0 and 1 | the chapter's description of a fraction smaller than one whole | printed in §8.1, p.185, in the table and in the closing blanks |
| greater than 1 | the chapter's description of a number larger than one whole | printed in §8.1, p.185 |
| area | the surface a rectangle covers, unchanged when its two sides are swapped | printed in §8.1, p.186 |
| length and breadth | the two sides of the rectangle that are interchanged | printed in §8.1, p.186 |
| order of multiplication | which factor is written first | printed as the block heading in §8.1, p.186 |
| Brahmagupta | the mathematician whose formula the chapter points to as a second reason the order does not matter | printed in §8.1, p.186 |
| scaling | stretching or shrinking a quantity by a factor | the explanation's word for what §8.1 pp.184–186 describes; not printed in this chapter |
| commutative | the property that the order of the two factors does not affect the product | an added term; not printed in this chapter, which states the property in words instead |
Where people slip up
- "Multiplication makes things bigger." The chapter's whole block exists to break this. Do not correct it by replacing it with "multiplying by a fraction makes things smaller", which is the same mistake with a different sign.
- "If one factor is a fraction, the product is below both numbers."
1/4 × 8 = 2is printed on p.184 precisely to stop this. The product is above the 1/4. - "Fraction means below 1." 4/3 is a fraction and sits above 1, and the chapter puts it in Situation 1 for that reason. The hinge is 1, not the presence of a fraction bar.
- "The conclusion is that a product always falls below both factors." The chapter's conclusion is per factor — a factor below 1 pushes the product below the other number. Below both is the special case where both factors are below 1, which is Situation 2 and only Situation 2. The chapter reaches that conclusion for one worked pair on p.185; it does not generalise it.
- "You can compare 6/20 with 3/4 by looking at them." You cannot, reliably, and the chapter does the rewriting to twentieths rather than asking the reader to see it. Show that rewriting as a step, not as a formality.
- "1 belongs in the table too." The block removes 1 in its opening sentence, because a factor of 1 leaves the other number exactly where it was and would make every row need an "or equal to".
- "Swapping the factors gives a different answer because you cut differently." You do cut differently — the two pictures on p.186 are genuinely different pictures — and you land on the same area. That is the point worth slowing down on.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 4 Q9
Transcript1,269 words
Does multiplying make a number bigger? Three times five is fifteen. Fifteen is bigger than three, and bigger than five. So the answer looks obvious, and as long as you stay with counting numbers it really is. But watch what happens when one of the two numbers is a fraction. A quarter times eight is two. Two is smaller than eight. Multiplying made the number smaller. So that question does not have an answer. It has a missing condition.
Today is about finding the condition, and it turns out to be a single number. Before anything else, one number has to be lifted out of the discussion. One. Multiply anything at all by one and it does not move. Twelve times one is twelve. Three quarters times one is three quarters. I checked that across every number I tried, and not one of them shifted by anything. So one is not interesting as a factor. It is the thing every other factor gets measured against.
Lift it out, and every other positive number is either above it or below it. That split is the whole answer, and I am going to make you wait for it. Back to a quarter times eight. The answer is two, and the interesting part is not the two. It is where the two sits. Two is well below eight. That is the part people find surprising. But two is far above a quarter.
So multiplying did not simply make the number smaller. It made it smaller than one of the two numbers and bigger than the other. That one line is enough to kill the first rule most people reach for. It is simply not true that having a fraction in the multiplication drags the answer below both of the numbers. Now take a case where both of the numbers are below one.
Three quarters times two fifths. Three twos are six. Four fives are twenty. Six twentieths. And now the awkward question. Is six twentieths bigger or smaller than three quarters? Look at those two as they are written and you genuinely cannot tell. Six is bigger than three. Twenty is bigger than four. Comparing the numerals gets you nowhere, so we do the honest thing instead. Rewrite all three numbers over the same denominator.
Three quarters is fifteen twentieths. Two fifths is eight twentieths. And the product is six twentieths. Now every one of them is a count of the same sized piece, so the numerators alone settle it. Fifteen. Eight. Six. Six is the smallest, so the product landed below both of the numbers that made it. That rewriting was not a formality, and it is worth doing slowly. Without it the numerals point in exactly the wrong direction.
Here is where the obvious rule breaks down completely. Take four thirds, and multiply it by four. Four thirds is a fraction. It has a bar, and a number above and below it. But four thirds is bigger than one. Four thirds times four is sixteen thirds, which is five and a third. That is above four, and it is above four thirds. The product went up, and one of the two factors was a fraction.
So the hinge was never the fraction bar. The hinge is one. Which leaves three situations to sort every possible pair into, and only three. Both numbers above one. The product lands above both of them. Both numbers below one. The product lands below both of them. One above and one below. The product lands somewhere between the two. Every example so far sits in one of those three rows.
Three times five: both above one, and fifteen is above both. Three quarters times two fifths: both below one, and the product is below both. A quarter times eight: one of each, and the answer landed in between them. Now I want you to state the rule, and I want you to state it carefully. Here is the shape of it, with one word left out. Multiply a number by a factor between zero and one, and the product is blank than that number.
Less. Multiply a number by a factor greater than one, and the product is blank than that number. Greater. Now notice what each of those two sentences is comparing. Each compares the product with the other number. One factor at a time. That phrasing is doing real work, and the next two minutes are about why. The tempting version of the rule is shorter, and it is wrong. It goes like this: multiply by a fraction and the answer comes out below both of the numbers.
A quarter times eight is two, and two is not below a quarter. So that version is dead on arrival, and it dies on an example we have already done. The correct statement is about one factor and the other number, never about both at once. Below both is a special case. It happens when both of the factors are below one. And when exactly one of them is below one, the product has to land between the two.
It is pushed down from the larger number and up from the smaller one at the same time. I did not want to take any of that on trust, so I swept it. Eight thousand two hundred and eighty-one pairs of positive numbers, each one multiplied out and compared against both of its factors. On four thousand and ninety-five of them a factor was above one, and every single time the product came out above the other number.
On another four thousand and ninety-five a factor was below one, and every single time the product came out below the other number. On ninety-one the factor was exactly one, and the other number never moved at all. Counter-examples: none. Two thousand and twenty-five products landed below both of their factors, and in every one of those, both factors were below one. And the tempting rule? Out of eight thousand one hundred and thirty-seven pairs with a fraction in them, it gets six thousand one hundred and twelve wrong.
One more thing, and it is a question that ought to bother you. Does it matter which of the two numbers you write first? A half times a quarter. Or a quarter times a half. Draw them both, side by side, and they really are two genuinely different pictures. One rectangle is half a unit wide and a quarter of a unit tall. The other is a quarter wide and half a unit tall. Different shape, different cuts.
But it is the same rectangle turned on its side, and turning a rectangle does not change its area. One eighth, both times. Across all eight thousand two hundred and eighty-one pairs, swapping never once changed the answer. Which means you can now answer a question without doing the multiplication at all. Five hundred and sixty-five over four hundred and sixty-five, times seven hundred and seven over six hundred and seventy-six.
Nobody in their right mind wants to multiply those two out. You do not have to. Put each factor against one instead. Five hundred and sixty-five is bigger than four hundred and sixty-five, so the first factor is above one. Seven hundred and seven is bigger than six hundred and seventy-six, so the second one is above one as well. Both above one. So the product is above both of them, and above one, and I never multiplied anything.
Multiplication scales. Whether it stretches or shrinks is decided by one, and you can see which before you calculate a thing.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- A fraction of a fraction, and why the numerators and denominators multiplyClass 7 · Ch 8, Working with Fractions
Comes up again in
- Why dividing can make a number biggerClass 7 · Ch 8, Working with Fractions
- Why multiplying by a decimal below 1 shrinks a numberClass 7 · Ch 4, Another Peek Beyond the Point
Either side of this one
- Cancelling common factors before multiplying, not afterClass 7 · Ch 8, Working with Fractions
- Restating a division as a missing-factor multiplicationClass 7 · Ch 8, Working with Fractions