Solving problems in standard form (Higher)

The nearest star to our Sun sits about four times ten to the power sixteen metres away. That is a number nobody can picture. So astronomers do not measure it in metres. They measure it in light years. Light travels three times ten to the power eight metres every second. A year is about three point two times ten to the power seven seconds. So one light year is nine point six times ten to the power fifteen metres. Divide the star's distance by that, and the answer is about four point two. Proxima Centauri, the nearest star beyond our own Sun, is four point two light years away. The light reaching Earth tonight left that star just over four years ago. The dividing was not the hard part. Deciding to divide was. In this video you will read a problem, choose the operation the words are asking for, and finish your answer in standard form.

Start with the decision that comes before any arithmetic.

A factory produced four million, two hundred and fifty thousand items last year. This year it produced six point one times ten to the power six items. Work out the total number of items produced over the two years, giving your answer in standard form.

Two numbers, written two different ways. Before you touch either of them, read the words. Total number produced over the two years. Total means everything counted together. That is an addition.

Here is how one examiner report described this exact situation. Most students added six million, seven hundred and eighty thousand to eight point two times ten to the power five, giving an answer of seven million, six hundred thousand thus gaining the first mark. Generally, students went on to give an answer in standard form. A common error was to subtract six million, seven hundred and eighty thousand from eight point two times ten to the power five.

The reason for that error is presentation. Six point one times ten to the power six looks like a special object. It is not. It is six million, one hundred thousand items sitting in a warehouse. How a number is written never changes what the situation asks you to do with it.

So add. Four million, two hundred and fifty thousand plus six million, one hundred thousand is ten million, three hundred and fifty thousand. Then the last line of the question asks for standard form. One digit before the point. One point zero three five times ten to the power seven.

Line one to carry into the exam hall. Words choose the operation.

Your turn, and this one is quick. A warehouse holds two hundred and fifty thousand boxes. A delivery of six point eight times ten to the power five boxes arrives. Which calculation gives the new total?

A. Six point eight times ten to the power five minus two hundred and fifty thousand. B. Two hundred and fifty thousand plus six point eight times ten to the power five. C. Two hundred and fifty thousand multiplied by six point eight times ten to the power five.

Have a think. I'll wait.

The answer is B. A delivery arrives, so the warehouse ends up with more boxes than it started with. More means add.

A subtracts, which is the error the examiner report named. C multiplies, which would mean two hundred and fifty thousand lots of six hundred and eighty thousand boxes. The words only said a delivery arrives, so neither of those fits.

Now a harder read. This time both numbers are in standard form, and the words never say which operation to use.

Light from the Sun travels at three times ten to the power eight metres per second. The Sun is one point five times ten to the power eleven metres from Earth. Work out how long the light takes to reach Earth, in seconds.

Nothing there says add, or times. So look at what you have and what you want. You have a distance in metres, and a speed in metres per second. You want a time in seconds. Metres per second means metres for every second, so metres divided by metres per second leaves seconds behind.

There is your operation. Divide the distance by the speed. One point five times ten to the power eleven, divided by three times ten to the power eight.

Have a go at this one. I'll wait.

The answer is five hundred seconds. The front numbers: one point five divided by three is nought point five. The powers: eleven take away eight is three. Nought point five times ten cubed is five hundred.

And look at what the question asked for. In seconds. It did not ask for standard form, so five hundred is the finished answer. The form your answer takes is set by the question, not by the numbers you started with.

Five hundred seconds is a little over eight minutes. So the sunlight on your face is eight minutes old.

Line two. When the words are quiet, the units confirm it.

Last problem. This one has two steps, and the second step is a conversion.

The mass of a planet is five point four times ten to the power twenty four kilograms. A moon has a mass equal to two percent of the planet's mass. Work out the mass of the moon, giving your answer in standard form.

Pause here and work it through. I'll wait.

The answer is one point zero eight times ten to the power twenty three kilograms. Here is the route to it.

Two percent is nought point nought two, so multiply. Front numbers: five point four times nought point nought two is nought point one zero eight. The power of ten is untouched. That gives nought point one zero eight times ten to the power twenty four.

A lot of answers stop right there. The calculation is done, so it feels finished. Examiners put it in one line, picked up part-way through: managed the first step but then could not convert their answer into standard form.

Nought point one zero eight is not between one and ten. One digit before the point. Move the point one place to one point zero eight, and the power drops by one, from twenty four to twenty three. One point zero eight times ten to the power twenty three kilograms.

Two separate jobs on one question. Do the calculation. Then convert. Line three. Standard form finishes it.

Now the exam side of this.

One practical habit. When a problem mixes standard form with an ordinary number, write both the same way before you calculate. Both ordinary, or both in standard form. Mixed notation on the page is where slips happen. The last line of a question is an instruction, not decoration. Giving your answer in standard form. In seconds. To three significant figures. Read it first, so you already know what your final line has to look like. And write the unit on it. Then a size check, which takes about five seconds. Two percent of something is far smaller than the something, so the power of ten should have come down, not gone up. Twenty four to twenty three. If your power moved the wrong way, go back through the working.

So, three problems, one routine. Here it is.

Read the words before you calculate. A number in standard form is still just a number, and how it is written never changes what the situation asks of it. When the words are quiet, let the units talk. Metres divided by metres per second gives seconds. Then check the last line of the question. If it says standard form, converting back is a separate job from the calculation. Finish it.

So. Words choose the operation. Units confirm it. Standard form finishes it.

That is the last line this chapter adds. Powers, roots, index laws, standard form. Each video left you one sentence to carry, and the one that did most of the work here came from the video on converting to and from standard form. One digit before the point.

And that completes our chapter on powers, roots and standard form. If you are following the course, the next chapter is fractions.

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