Hello vergita,
1. A factor is defined as: “One of two or more quantities that
divides a given quantity without a remainder. For example, 2 and 3 are
factors of 6; a and b are factors of ab.” Or as “any of the numbers
(or symbols) that form a product when multiplied together” or “one of
two or more integers that can be exactly divided into another integer”
http://dictionary.reference.com/search?q=factor&r=67
So, you need to find which numbers can be multiplied by each other to
give the numbers in the question.
26 – the only numbers you can multiply together to get 26 are 13 x 2
and 1 x 26. Therefore, the factors of 26 are 1, 2, 13 and 26.
30 – you can get 30 by multiplying 10 x 3 or 1 x 30 or 6 x 5.
However, you can get 10 by multiplying 5 x 2. So all the factors of
30 are 1, 2, 3, 5, 6, 10 and 30.
(BTW, because you might need this in future: if you want to have the
“prime factors” of a number, these are the factors which cannot be
divided by another whole number. So the prime factors of 30 are 5, 3
and 2. )
52 – you can get this by multiplying 1 x 52, 26 x 2, and also by
multiplying 13 x 4. Moreover, you can get 26 by multiplying 13 x 2,
and 4 by multiplying 2 x 2. So all the factors of 52 are 1, 2, 4, 13,
26 and 52.
(and the prime factors are 2 and 13).
65 - you can get 65 by multiplying 5 x 13 and 1 x 65. No other whole
numbers can be multiplied to get 65, so the factors of 65 are 1, 5, 13
and 65.
84 - you can get 84 by multiplying 1 x 84, 2 x 42, or 4 x 21, but you
can get 21 by multiplying 7 x 3, and 42 by multiplying 6 x 7, and 6 by
multiplying 2 x 3. So all the factors of 84 are 1, 2, 3, 4, 6, 7, 21,
42, 84.
2. The lowest common multiple is the first number you find in the
multiplication tables of all the numbers with which you are dealing.
It’s easier to show this by working out your question rather than
trying to explain in words.
You need to find the lowest common multiple of 9, 2, and 11
Start by writing out the 9 times table: 9, 18
And the 2 times table: 2, 4, 6, 8, 10, 12, 14, 16, 18
OK, you’ve found that 18 is the first number that belongs to both
sequences, so that is the lowest common multiple of 9 and 2. However,
18 does not belong in the 11 times table, so that is not the answer.
However, did you notice that 18 equals 9 x 2? This hints at a short
cut that you can use. Since no number will divide into both 2 and 9,
except for 1, their lowest common multiple is found by multiplying
them by each other.
No one number will divide into 2, 9 and 11. So, the lowest common
multiple for these three numbers is found by multiplying all three: 2
x 9 x 11 = 198.
You can prove this to yourself by writing out all the multiplication
tables for 2, 9, and 11 and seeing which number first appears in all
three. It will be 198.
The quickest way to do this is to take the highest number you have,
and write out its multiples until you find a number that can be
divided by each of the smaller numbers.
In your example, we need to take multiples of 11: 11, 22 (can be
divided by 2 but not by 9), 33, 44 (by 2 not 9), 55, 66 (by 2 not 9),
77, 88 (by 2 not 9), 99 (by 9 not 2), 110 (by 2 not 9), 121, 132 (by 2
not 9), 143, 154 (by 2 not 9), 165, 176 (by 2 not 9), 187, 198 (by 2
AND by 9).
You can find some other examples here:
http://www.themathpage.com/ARITH/compare-fractions.htm and here
www.schoolnetindia.com/nlc/NLCLessons/lessons/factors.ppt
3. The highest common factor is “the largest number that divides
evenly into each of a given set of numbers”
http://www.bartleby.com/61/96/G0249600.html
The way to solve this is to look at the smallest number you have in
the series, see how it can be produced by multiplying various pairs of
numbers, and then see which of these is the highest number than can
also be multiplied to produce the other numbers in your series.
a.)12, 72, 60
Looking at 12, this can be produced by multiplying 1 x 12, 3 x 4, or 6
x 2. The largest number here is 6. 6 goes into 72 (12 x 6) and also
into 60 (10 x 6). Therefore, 6 is the highest common factor
b.)15, 75, 45
Looking at 15, this can be produced by multiplying 3 x 5 or 1 x 15.
The highest number is 15. 75 can be obtained by multiplying 5 x 15,
and 45 can be obtained by multiplying 3x 15. Therefore, the highest
common factor is 15.
c.) 48, 72
Looking at 48, this can be produced by multiplying 6 x 8, 3 x 16, 4 x
12, 2 x 24, and 1 x 48. The highest number is 48. However, 48 does
not go into 72. The next highest number is 24. You can obtain 72 by
multiplying 3 x 24. Therefore, the highest common factor is 24.
4. Convert fractions into decimals. It is useful to know the
relationship between some fractions and decimals: for example 1/2
(half) = 0.5, 1/4 (quarter) = 0.25, 1/5 (fifth) = 0.2, 1/10 (tenth) =
0.1, etc.
a.)17/5 Divide 5 into 17. This gives 3, with 2 left over, i.e. it
gives 3 and 2/5. One fifth is equivalent to 0.2, 2/5 are equivalent
to 0.4. Therefore the answer is 3.4
b.)21/6 Divide 6 into 21. This gives 3, with 3 left over, i.e. it
gives 3 and 3/6. However, 3/6 is the same as one half (1/2) when you
divide top and bottom of 3/6 by 3. One half in decimals is 0.5.
Therefore the answer is 3.5.
c.)33/12 Divide 12 into 33. This gives 2 with 9 left over, i.e, it
gives 2 and 9/12. However, you can divide the top and bottom of 9/12
by 3, which gives you 3/4. One quarter in decimals is 0.25, so three
quarters are 0.75. Therefore the answer is 2.75.
5. To convert decimals into fractions, you change the numbers after
the decimal point first into tenths, hundredths, etc, and then break
them down further if possible.
a.) 0.423 - There are three numbers after the decimal point, so
we are dealing with thousandths:
0.423 = 423/1000. There is no number that can be divided into 423 and
into 1000, so the answer is 423/1000.
b.)1.02 - Look just at the decimals (.02). There are two numbers
after the decimal point so we are dealing with hundredths. 2/100 can
be divided by 2 above and below to give 1/50. Therefore the answer is
1 and 1/50, or 51/50.
c.) 0.53 - Again, there are two numbers after the decimal point so we
are dealing with hundredths. This gives 53/100. There is no number
that can be divided into 53 and 100, so the answer is 53/100.
d.) 0.50 You should be able to recognise this one as being
equivalent to 1/2. However, if you don’t, you can use the same
method: There are two numbers after the decimal point so we are
dealing with hundredths. 50/100 can be divided by 10 above and below
to give ½.
e). You have given this as 0.50, same as (d), so the answer is the
same. However, if you made a typo, try to figure it out for yourself,
but do feel free to get back to me through a request for clarification
if you get stuck.
f.) 0.25 You should be able to recognise this one as being
equivalent to 1/4. However, if you don’t, you can use the same
method: There are two numbers after the decimal point so we are
dealing with hundredths. 25/100 can be divided by 25 above and below
to give 1/4.
g). 0.125 – Can you see that this is half of 0.25? If yes, then you
can say straight away that it is equivalent to 1/8. If not, use the
same method as above. There are three places after the decimal point,
so we are dealing with thousandths. This gives 125/1000. 5 can go
into these numbers above and below to give 25/200, and 25 can go into
this to give 1/8 (I broke the division up into two stages to make it
simpler).
6). The trousers cost $25 and the shirt cost $21, making a total of
$46.
If these prices were 20% off, that means Sina only paid 80% of the
total.
80% is equivalent to 80/100.
Trousers: 80/100 is $25. To get 100%, ie the original price, you need
to find another 20/100. But 20/100 is one quarter of 80/100. So, a
quick way to do this is to calculate one quarter of 25 and add it on
to 25. One quarter of 25 = 25/4 = 6 and 1/4, but 1/4 is 0.25 in
decimals, so one quarter of 25 = 6.25. Therefore, the original price
was $31.25.
The quick way to do this is to invert the fraction:
80/100 = 25. 100/100 = 25 x 100/80 = 2500/80 = 250/8 = 31.25
Shirt: using the quick way:
80/100 = 21. 100/100 = 21 x 100/80 = 2100/80 = 210/8 = 26.25
Therefore the shirt originally cost $26.25.
7. The oven installation cost $54 plus VAT, and VAT is 17.5%.
So we need to find 17.5% of $54 and add it to the price.
Again, you can do this a quick way. If 100% is the price, then the
price including VAT is 117.5%.
100/100 = 54
117.5/100 = 54 x 117.5/100 (but 117.5/100 = 1.175 when you divide top
and bottom by 100). Therefore you can calculate is as 54 x 1.175 =
63.45
The price with VAT = $63.45.
OK, I used a calculator for the last step. If you need to do the
calculation in your head, you can break it down further because 54 = 6
x 9.
Therefore, you can multiply 1.175 first by 6, to get 7.050 and then
multiply this by 9.
I hope this helps, but please request clarification if something |
