PrepShorts · Study sheet · Class 7 Mathematics · Chapter 3, A Peek Beyond the Point
Chapter 3 · A Peek Beyond the Point
When a decimal-looking number is not a decimal: 4.5 hours, 5.5 overs
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4.5 hours is four and a half. 5.5 overs is not five and a half. The digit after the point does not always count tenths.
The idea
A decimal point means "tenths to the right" only because the unit was agreed to be cut into ten. Where the world cuts the unit some other way — sixty minutes to an hour, six balls to an over, twelve inches to a foot — the same printed mark carries a different agreement, and reading it as a decimal is not a slip of arithmetic but a mistake about which convention is in force. The chapter's cricket case is the sharpest evidence: the careless reader gets the right number of balls for entirely the wrong reason, and a careful decimal reading gets a different answer (Part I, §3.8, p.78).
What you should be able to do
- Say what a digit after the point counts in a genuine decimal number
- Work out, for a quantity written in a non-decimal convention, what one tenth of its unit would be worth, and compare that with the convention actually used
- Convert a duration written in decimal hours into hours and minutes
- Convert a length written in decimal feet into feet and inches
- Explain what an over-count such as 5.5 means, and which denominator is in force
- Identify, from an everyday quantity, whether the notation after the point is decimal or not, and say how you can tell
- Give a further example of a decimal-looking number that is not one
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| non-decimals | the chapter's own word for quantities written with a point but not in tenths | printed in single quotes in Part I, §3.8, p.78 |
| decimal-looking | the chapter's description of such a written form | printed in Part I, §3.8, p.78 |
| decimal point | the mark that fixes where the units place ends | printed in Part I, §3.4, p.62 |
| tenths | what the first place after a genuine decimal point counts | printed in Part I, §3.2, p.48 |
| over | in cricket, a unit of six balls | printed in Part I, §3.8, p.78 |
| ball | one delivery; six of them fill an over | printed in Part I, §3.8, p.78 |
| inch | a unit of length, twelve of which fill a ft | printed in Part I, §3.8, p.77 |
| ft | the chapter's abbreviation for the length unit twelve inches fill | printed in Part I, §3.8, p.77 |
| denominator | the number of equal parts a whole is cut into | printed in Part I, §3.8, p.78 |
| mixed unit notation | writing a quantity as a whole part and a count of some non-decimal sub-unit | an added term; not printed in this chapter, which shows the forms without naming the practice |
| sexagesimal | counting in sixties, as a clock does | an added term, not printed in this chapter; offered only if the explanation wants a name for what a clock does |
Where people slip up
- "4.5 hours means four hours and five minutes." It is the first candidate the chapter rejects, and it is the commonest reading among students who see the digits and think of a clock face.
- "4.5 hours means four hours and fifty minutes." The second rejected candidate, and it comes from treating the digit after the point as a count of tens of minutes.
- "2.5 ft means 2 ft 5 inches." The whole point of the door story. Five inches is not half a ft, and the two notations differ by an inch.
- "5.5 overs means five overs and five tenths of an over." In cricket the over is cut into six, so the second digit counts balls out of six, not tenths.
- "So the careless reading works for cricket." It gives the right number of balls here by coincidence, not by reason, and the same carelessness gives the wrong answer for the bus and for the door. Show why it works and then show that it does not generalise, rather than leaving a student with a rule that happens to have worked once.
- "A number with a point is always a decimal number." The section exists to refuse this. What makes a number decimal is that the unit was cut into ten, and that is a fact about the situation, not about the printed mark.
- "You can add times as if they were decimals." Two durations written as 2.45 and 1.30 in the clock sense do not add like decimals, because the sub-unit does not carry at ten. This case is an added extension of the chapter's argument and should be flagged as such.
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Worked answers to this chapter’s exercises
Transcript1,260 words
A message arrives. The bus reaches the station four point five hours after midday. So what time should you be standing there? Three answers suggest themselves, and one of them is probably already in your head. Five past four, because the five after the point looks like a count of minutes. Or ten to five, four fifty, because that five might be five tens of minutes instead. Or twenty five past four, which is the one people arrive at last and defend hardest.
And all three of them are wrong. And the nearest of the three is still five minutes out, which is quite enough to miss a bus. But what has gone wrong here is not arithmetic. Nobody has added anything up incorrectly. So look at what the message actually says, one piece at a time. Four point five hours. Not four hours and something. Four hours, and then five tenths of an hour.
Because that is what a decimal point declares, every single time. The digit immediately after it counts tenths of whatever unit has been named. Not minutes. Not tens of minutes. Tenths of an hour. So before you can answer at all, you have to know one thing. How long, exactly, is one tenth of an hour? And notice that this is a question about hours. It is not a question about the notation.
Here is an hour, drawn as a bar. Cut it into ten equal parts, because ten equal parts is exactly what the point asked for. Now, how many minutes is one of those parts? An hour is sixty minutes, and sixty divided by ten is six. One tenth of an hour is six minutes. Not the six on the clock face. Six minutes of time. And notice something quietly lucky about that number.
It came out whole. Ten goes into sixty exactly, so every tenth of an hour lands on a minute. A seventh of an hour would not. Decimal hours work at all only because sixty happens to be friendly to ten. Now take five of those parts, because the message said five tenths. Six minutes, six, six, six and six. Thirty minutes in all. Which you could have reached a shorter way. Five tenths is a half, and half an hour is thirty minutes.
So the bus arrives four hours and thirty minutes after midday. Half past four. Not five past. Not ten to five. Not twenty five past. Four point five hours is four and a half hours, and it always was. All three of the tempting answers came from reading that five as something a clock counts. Which is worth understanding properly, because the temptation is not stupidity. Here is the hour again, cut into ten.
And here it is cut into sixty, which is how a clock cuts it. Lay one over the other and look at where the lines fall. They do line up. Every tenth mark of the first grid sits neatly on a mark of the second. But the two grids have different numbers of pieces, so a count on one is not a count on the other. Five out of ten is not five out of sixty.
One of them is half the hour. The other one is five minutes. And reading a decimal point on a clock is reading a count off one grid against the other. The same thing again, in wood, with a hole in a wall. Someone measures an opening for a new door. It comes to two feet and five inches. She telephones a carpenter and says: make it two point five feet.
The carpenter builds a door two point five feet wide, exactly as he was asked. Twelve inches make a foot. So half a foot is six inches. Which means the door he builds is two feet and six inches. And the opening is two feet and five inches. So the door is one inch too wide. It does not shut. It is never going to shut. So who made the mistake here? It is worth being exact about this.
Not the carpenter. He read two point five feet correctly and built precisely that. Not the measurement either. Two feet five inches was right when she took it. The mistake happened in between, in a single sentence spoken over a telephone. She had a length written in feet and inches, and she read it aloud as though it were a decimal. Two feet five inches became two point five feet, and those are two different lengths.
One inch apart, which sounds like nothing until a door has to pass through a frame. Two people used one notation, and each of them meant something perfectly sensible by it. That is the entire failure. Nobody was careless, and nobody was bad at sums. Now cricket, which is where this becomes genuinely interesting. A scoreboard reads: overs left, five point five. An over is six balls. Six, not ten.
So how many balls are left? Two answers are on offer. Five overs and five balls, or five overs and three balls. The correct answer is five overs and five balls. Because in cricket that point does not mean tenths at all. It means balls, out of six. Five point five overs is five and five sixths of an over. Thirty five balls left to bowl. And now the trap, which is why this example is worth more than the other two put together.
Suppose you had ignored the point completely. Suppose you had simply looked at that second five and said: five balls. You would be right. Exactly right, and for no good reason whatsoever. Because take the very same habit to the bus, and it hands you five past four. Right once, wrong twice. That is not a method. That is luck. And here is the other half of the trap. A careful decimal reading of five point five overs gives five tenths of six balls, which is three.
Three. Which is exactly the wrong answer that was sitting on offer beside the right one. So here is what all three of these have in common. The very same mark, meaning three completely different things. Point five of an hour is thirty minutes. Point five of a foot is six inches. Point five of an over is five balls. Thirty, six and five, out of one identical piece of notation.
And there is no way at all to tell them apart by looking at it. A point means tenths only because somebody agreed to cut that unit into ten. So what makes a number a decimal is not the point. It is the unit. Which leaves a question worth carrying around with you. Where else in the world does this happen? Clock times, lengths in feet and inches, and sports scores are three you have just met.
Here are three more, and see whether you can go on adding to them. Adding times as though they were decimals. Two forty five plus one thirty is not three point seven five hours. It is four hours and fifteen minutes, because minutes carry at sixty and decimals carry at ten. Version numbers. Three point ten comes after three point nine, which as decimals would be backwards. And in baseball, five point two innings means five innings and two outs, out of three.
Every one of those is a number with a point in it, and not one of them is a decimal. Look at the unit, never at the mark.
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
- Extending Indian place value to the right of the pointClass 7 · Ch 3, A Peek Beyond the Point
- Why measurement units are built in tensClass 7 · Ch 3, A Peek Beyond the Point
Either side of this one
- What a misplaced decimal point costsClass 7 · Ch 3, A Peek Beyond the Point
- Rival ways of splitting off the fractional part, and why the point wonClass 7 · Ch 3, A Peek Beyond the Point