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Answer:

#13 7/24#

Explanation:

#"A "color(blue)"mixed number"# is made up of a whole number and a fraction, being added to it.

One way of adding them is to ADD the whole number parts together and the fraction parts together and combine the 2 results.

#rArr4 1/8+3 2/3+5 1/2#

#=4+1/8+3+2/3+5+1/2#

#=4+3+5+(1/8+2/3+1/2)#

#=12+(1/8+2/3+1/2)larrcolor(red)" whole numbers added"#

Before adding the fractions, we require them to have a #color(blue)"common denominator".# That is the same value on the denominator of each fraction.

This can be achieved by finding the #color(blue)"lowest common multiple"#
( LCM) of 8 , 3 and 2

The LCM of 8 , 3 and 2 is 24

We can change each fraction to this denominator.

#(1/8xx3/3)+(2/3xx8/8)+(1/2xx12/12)#

#=3/24+16/24+12/24#

Now they have a common denominator, we can add the numerators, leaving the denominator.

#rArr(3+16+12)/24=31/24=1 7/24larrcolor(red)" fractions added"#

Combing the 2 additions.

#rArr12+1 7/24=12+1+7/24=13 7/24#

Answer:

For a while, explicitly put the -1

Explanation:

Let us agree on this #-5=(5)(-1)#

Anything multiplied by 1 is itself, and anything multiplied by (-1) is its opposite.

Let us agree on this #5=(-1)(-1)(5)#

A positive number multiplied by (-1) results in that negative number, and a negative number multiplied by (-1) results in that positive number.

A positive number multiplied by (-1) an even number of times results in that positive number (no change).

A negative number multiplied by (-1) an even number of times results in that negative number (no change).

Multiplying by (-1) twice undoes the multiplication.

So, with this in mind, let us consider addition.

#-2+3#

We can write this as

#(-1)(2)+3#

Addition has a property that allows us to do it in any order, and still get the same result. It's called the communitive property.

#a+b=b+a#

Well, we just turned our problem into an addition problem. That means we can rearrange the terms. So, let's do that:

#3+(-1)(2)#

According to order of operations, we must multiply before adding, so let's multiply that (-1):

#3-2#

So, it appears that an addition problem with a negative in front is really a subtraction problem in disguise.

Let's try another one:

#-2-3#

We will again replace the negatives with (-1), but it is important to remember that we are adding. We always add, but sometimes we add negative numbers.

#(-1)(2)+(-1)(3)#

Both terms are being multiplied by (-1), which brings us to another property. The distributive property says:

#ab+ac=a(b+c)#

Let us pull out the (-1) in like fashion.

#(-1)(2+3)=(-1)(5)=-5#

Finally, we have

#2-3#

Again, this can be thought of as:

#2+(-1)(3)#

Move the bigger number to the front

#(-1)(3)+2#

Multiply first

#-3+2#

We can pull out a (-1) here too because anything multiplied by 1 is itself and anything multiplied by (-1) is its opposite, so positive becomes negative in that case.

#(-1)(3-2)#

#=(-1)(1)#

#=(-1)#

Now, it would be silly to do all of this every time. You will very rapidly internalize these ideas, but hopefully this will help in thinking about it.

We have covered addition, multiplication, and subtraction. Division might be a little tricky, but I know you can get it.

#-2/3=(-2)/3=2/-3#

Let us see them more clearly:

#(-1)2/3=((-1)2)/3=2/((-1)3)#

Remember that an even number of (-1) produces no change.

#2/3=((-1)2)/((-1)3)=((cancel(-1))2)/((cancel(-1))3)#

So, with this knowledge, let's solve an equation.

#(-5+5-2+7*-2)/-2#

#=((-1)5+5+(-1)2+7*(-1)2)/((-1)2)#

#=((-1)(5-5)+(-1)2+(-1)14)/((-1)2)#

#=((cancel(-1))(0)+(cancel(-1))2+(cancel(-1))14)/((cancel(-1))2)#
Note:All terms contain (-1), so it is a common factor.

#=((0)+2+14)/(2)#

#=(2+14)/2#

#=16/2#

#=8#

Get the hang of it, and then abandon it. It will just be automatic.

Answer:

We can work backwards and use reciprocal of the fraction

Explanation:

The reciprocal of the fraction is just the flipped version of that fraction. So, #1/2# becomes 2. And #1/4# becomes 4.

A fraction multiplied by the reciprocal is always 1.

#1/2*2/1=2/2=1#

The question gives a specific value of 60 cents as a result of spending #1/4# of her money after spending #1/2# of her money.

So, if we take the reciprocal of the fractions, we can put the money back. For example, we go from some amount to 60 cents, by spending #1/4# of the money. Let's give the money back.

#60*4=240#

Now, before this she spent half the money, so again we can take the reciprocal, which is 2.

#240*2=480#

She started with 480 cents. However, we might want to express this as dollars. So, we can divide by 100 because there are 100 pennies in a dollar.

#480/100=4.80#

Pam started with $4.80.

We want to double check though since we might be wrong. Let's work through the problem forward and see if we get 60 cents again.

First, use the value that is in cents because the answer is in cents: 480.

Next, she spent half the money:

#480*(1/2)=240#

Now, she spent a fourth of the money.

#240*(1/4)=60#

And this checks out. So, we can be sure that Pam started with $4.80.

Answer:

They are not equal, but their values are very close.

#pi != 22/7, " "pi ~~ 22/7#

Explanation:

#22/7 and pi# are not equal, but we can say that

#pi ~~ 22/7" "# they are approximately equal.

#pi# is an irrational number - it is an infinite, non-recurring decimal.
Its value is #3.141592654..............# There is NO pattern.

#22/7# is a fraction which is very close to #pi#.

However, #22/7# is a rational number which can be written as a recurring decimal . #22/7 = 3.142857bar142857...# There IS a pattern.

If #pi and 22/7# were equal, their difference would be #0#

However: #22/7 - pi = 0.00126448925...#

Therefore they are not equal, but,

#22/7 = 3 1/7# is very close to #pi#

Answer:

#LCM = 36#

Explanation:

Write each number as the product of its prime factors, then you know what you are working with.

Notice that you do not even need to consider #4#, because #4# is a factor of #12#, so any multiple of #12# will be a multiple of #4# as well.

#color(white)(............) 4 = 2xx2#
#color(white)(............) 9 = " "3xx3#
#color(white)(..........) 12 = 2xx2xx3#
#color(white)(.................) darrcolor(white)(.)darrcolor(white)(m)darrcolor(white)(.)darr#

#LCM =" " 2xx2xx3xx3 = 36#

Notice that in factor form:

#2xx2# is there for the #4#
#2xx2xx3# is there for the #12#
#3xx3# is there for the #9#

All the numbers are in the LCM, but there are no unnecessary factors.

Answer:

#31.5# litres

Explanation:

Find the total volume first:

#450# glasses each contain #350# ml

#450 xx 350 = 157,500# ml.

Convert to litres immediately (#div 1000#)

#157,500 div 1000 =157# litres

The ratio: #" water : juice "# is #" "4" : "1#

This means that #4/5# is water and only#1/5# is juice.

The volume of juice is therefore:

#1/5 xx 157 = 31.5# litres

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