I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.

Notes

As of 5 October: MATH6000 Lecture Notes (with gaps).

Also an E-Book: Engineering Mathematics by John Bird.

Also I don’t know why I didn’t advise this earlier — you would be well worth investing in a ring binder (to be kept at home or whatever) for your notes. You can see already the amount of sheets. You should be organised with these and only bring the ones we are working on to class.

Test!

Your first assessment is on this Wednesday 10 October in Week 4. The exact time and location depends on your class group:

Common Entry Science: 17:00 in the West Atrium

Biosciences:                              17:00 in the West Atrium

Computing:                               18:00 in the West Atrium

The test will be a 45 minute, multiple-choice test (15 questions equally-weighted questions), without negative marking and calculators will not be permitted.

The sample, which you should be very familiar with, may be found here.

Do not miss this first assessment. The policy on missed assessments is very strict in CIT as you can see here.

If you are really struggling with the sample assessment, I hope that you have been attending tutorials as these are your best chance of getting help. If you haven’t been attending tutorials please, please come to one early next week and also use the Academic Learning Centre if possible (see below).

If you are finding the lectures all a bit easy I warn you not be to complacent. As the test is non-calculator you will need to do long multiplication, long division, etc. You need to practise these. Please sit down and do the sample assessment before thinking that you will waltz in and get 100%.

For the rest of us in the middle we should find some questions straightforward and some questions a bit harder. I advice that you try exercises similar to the questions that you find a little harder, e.g. division of fractions.

In the test itself relax and take your time. I would advise that you do the questions you find easier first and then work on the harder questions. If you can’t get an answer right you know that you will guess the answer with probability 1/4.

Below find additional remarks and solutions to a number of the sample questions.

Question 2

Evaluate $\displaystyle 5\frac34+4\frac23-8\frac12$.

Remark: I recommend that you look at this question and think… mixed fractions are good for telling me the magnitude of a quantity but they are not much good for arithmetic. I should change them into fractions. So what you do is look at

$\displaystyle 5\frac{3}{4}$

and recognise this as $\displaystyle 5+\frac{3}{4}$. Now $5$ has $5\times 4=20$ quarters in it, plus the three in the fraction and we get

$\displaystyle 5\frac34=\frac{23}{4}$.

So it turns out it was “four times five plus three over four”. Now we add the fractions by putting them all over a common denominator.

Question 3

Find the value of $\displaystyle\frac{\frac23-\frac14}{\frac45+\frac13}$.

Remark: The first thing we should think is, euh, what a mess. I can at least tidy it up by finding the top and bottom separately.

Once this is done you are left with something equal to

$\displaystyle\frac{\frac{5}{12}}{\frac{17}{15}}$   (*)

There are two ways to proceed from here. One involves multiplying above and below by something that will cancel the nasty $12$ and $15$. Can you think of a number that might do this?

Otherwise what we do is note that when we write $\displaystyle\frac{4}{5}$ we also mean $4\div 5$. So that means that (*) is just

$\displaystyle\frac{5}{12}\div \frac{17}{15}$.

If you have done enough exercises you should be happy enough to say, ah, to divide by a fraction I invert and multiply. Multiplication of fractions is easier than addition: just multiply numerators by numerators and denominators by denominators.

If however we have not done division by fractions or are not happy with invert and multiply let us see why this is the correct way to do division by a fraction. Recall that for natural numbers we have

$\displaystyle 5\div 6=5\times\frac{1}{6}=\frac56$.

That is to divide by six, you ‘invert’ $\displaystyle6\rightarrow\frac16$ and multiply! Inverting a number $x$ really means finding it’s multiplicative inverse,  that number $x^{-1}$ such that

$x\times x^{-1}=1$,

morryah $\displaystyle x^{-1}=\frac{1}{x}$.

So to divide by $\displaystyle \frac{17}{15}$ we multiply by the inverse of $\displaystyle\frac{17}{15}=\frac{1}{\frac{17}{15}}$. We showed in lectures that the inverse of a fraction is found by ‘inverting’ the numerator and denominator:

$\displaystyle\frac{1}{\frac{17}{15}}=\frac{15}{17}$.

This could also be seen, here, by multiplying above and below by 15 AKA multiplying by one.

Question 4

Calculate  $2.6\times 0.016 - 0.0248$ correct to 2 significant figures.

Remark: To multiply together $2.6\times 0.016$ or any other pair of decimals for that matter, you are better off just multiplying the raw figures together: $26\times 16$. Do this by long multiplication and you get 416. Now we were asked to calculate $2.6\times 0.016$ not $26\times 16$. So look at this in another way:

$\displaystyle 2.6\times 0.016=\frac{26}{10}\times\frac{16}{1000}=\frac{26\times 16}{10000}$.

So to find $2.6\times 0.016$ just take your $26\times 16$ and divide by $10,000$. Now to divide by 10,000 you divide by ten once, twice, thrice, four times. Dividing by ten is equivalent to moving the decimal place one place to the left so we get:

$416\rightarrow 41.6\rightarrow 4.16\rightarrow 0.416\rightarrow 0.0416$.

Do we have to think of this everytime? Not at all!

Just take the raw figures, e.g. 26 and 16, and multiply them together. Now add up the number of decimal places in the original, e.g. four in $2.6\times0.016$ and take the $26\times 16=416$ and move back by four decimal places.

Question 6

Evaluate $s=ut+\frac12 at^2$ when $u=10.4$$a=-3.8$ and $t=5$.

Remark: Symbolically plugging in the values should be no problem:

$\displaystyle s=(10.4)(5)+\frac{1}{2}(-3.8)(t)^2$.

Just a pointer — the second term here is a positive number times a negative number times a positive number. It is going to be a negative number. Don’t mess around with multiplying positive by negative and such just get the minus out of there and write:

$\displaystyle s=(10.4)(5)-\frac{1}{2}(3.8)(5)^2$.

Question 9

If both sides of a rectangle are increased by 20%, what is the percentage increase in the area?

Solution 1 — Proper and Full Solution If we don’t know the dimensions of the rectangle we may as well denote its width and length by $w$ and $l$ respectively. So the original rectangle has an area of $l w$ Now to increase by 20% multiply by 1.20. I recommended multiplying by 1.20 to do an increase like this. The $1=100$% in 1.20 gives you the ‘original’ and the $0.20=20$% gives you the extra. The new width and length is $1.2w$ and $1.2l$:

The new area is $(1.2l)(1.2w)=1.44lw$: 44% more than the old area.

Solution 2 — Cheeky and Clever for Test: If we assume (which we shouldn’t really) that the test examiner is not going to make a mistake with the questions, we might infer from the question that it doesn’t appear to matter what the dimensions of the rectangle are: otherwise the examiner would have given us specific dimensions. In other words, it appears that whatever the dimensions of the rectangle are, the percentage increase in area caused by this change will be same. Therefore choose a nice rectangle, say a square of side length 10 which has area 100 (I knew there was a reason I told ye that all squares are rectangles!). Now increase the side lengths by 20%.

So the new area is 144: 44% more than the old area.

Question 12

A loss of 20% is made on the sale of an item. If the item was sold for €160, what was it bought for?

Remark: Recall the discussion on mark-up vs margin. In this test profit and loss will be profit and loss mark-up. This means that in this particular question €160 represents the original price less 20%. In other words €160 is 80% of the original price.

If an item was bought for €10 and sold for €12 and you are asked to ‘find the profit’. Please find the profit mark-up: what percentage of the cost price was added as profit:

profit mark up = $\displaystyle\frac{2}{10}=\frac{\Delta P}{P_0}$,

where $\Delta P$ is the profit and $P_0$ is the original, cost price.

If you want to find the margin (we don’t here), use

profit margin = $\displaystyle\frac{2}{12}=\frac{\Delta P}{P_1}$,

where $P_1$ is the new, selling price.

Exercises

Quite apart from the exercises in the notes, there are also exercise sheets on Blackboard.

Common Entry Science: By-and-large, next week we will work on the sample assessment on Tuesday but looking at Mensuration & Approximation on Thursday.

Biosciences: Your ‘other’ tutor and I may work with you on the sample assessment or the exercise sheets. I shall work with you on exercises in the notes.

Computing: Tim Buckley will work with you on the sample assessment or the exercise sheets. With the DCom group I might work with you on the sample assessment also. The DNet/ITM group will look at mensuration, approximation & statistics on Thursday. If you do not know what tutorial group you are in please consult here.

If you are comfortable with the material for the first assessment I invite you to look at the later sections: mensuration, approximation & statistics. There is another test in Week 6.