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2

Answer:

#140#

Explanation:

Use PEMDAS to help with the order of operations:

P for Parentheses

E for Exponents

MD for Multiplication and Division (left to right)

AS for Addition and Subtraction (left to right)

So we find:

#(4xx4(6+4)-3(8+1)+2)-:3+(5+2)(5+2)-2xx2+(42-:6)(42-:6)+1#

#=(4xx4(10)-3(9)+2)-:3+(7)(7)-2xx2+(7)(7)+1#

#=(4xx40-27+2)-:3+49-2xx2+49+1#

#=(160-27+2)-:3+49-2xx2+49+1#

#=(133+2)-:3+49-2xx2+49+1#

#=135-:3+49-2xx2+49+1#

#=45+49-4+49+1#

#=94-4+49+1#

#=90+49+1#

#=139+1#

#=140#

1

Answer:

The explanation is much longer than doing the mathematics.

The ratio of 15 boys to 10 girls is equivalent to the ratio of 3:2

Explanation:

Consider the starting point:
We have 3 boys and 2 girls. This gives a total count of 5

So we need to see how many lots of 5 will fit into 25.

5 lots of 5 gives 25

So we have 5 lots of the ratio 3:2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Method 1")#

5 lots of 3:2 #-> 5xx ( 3:2) =(5xx3) : (5xx2) = 15:10#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Method 2")#

Write the ratio in fractional form

#color(magenta)("We can do this as long as we do not view and treat it like a fraction")#

It does not matter in this case which we put on the top. I chose:

#("boys")/("girls") -> 3/2#

Multiply a value (or system) by 1 and you do not change the value. However, 1 comes in many forms.

#color(green)(3/2color(red)(xx1) " "->" "3/2color(red)(xx5/5))#

#" "=color(green)((3color(red)(xx5))/(2color(red)(xx5))#

#" "=15/10 =("boys")/("girls") #

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The ratio of 15 boys to 10 girls is equivalent to the ratio of 3:2

#15:10-=(15-:5):(10-:5)=3:2#

where #-=# means equivalent to

1

Answer:

#16 2/7 - 2 1/2*1 1/7 + 9/14 = 14 1/14#

Explanation:

Given:

#16 2/7 - 2 1/2*1 1/7 + 9/14#

First let us convert all of the mixed numbers to improper fractions:

#16 2/7 = 16 + 2/7 = (16*7)/7+2/7 = 112/7+2/7 = 114/7#

#2 1/2 = 2 + 1/2 = (2*2)/2 + 1/2 = 4/2+1/2 = 5/2#

#1 1/7 = 1 + 1/7 = 7/7+1/7 = 8/7#

So our original expression can be rewritten as:

#114/7 - 5/2*8/7 + 9/14#

Next note that multiplication has higher precedence than addition or multiplication, so we need to perform the multiplication #5/2*8/7# first:

#114/7 - color(blue)(5/2*8/7)+9/14 = 114/7 - (5*8)/(2*7)+9/14 = 114/7-40/14+9/14#

In order to add or subtract these fractions, they need to have the same denominators, so we multiply #114/7# by #2/2# to give it a denominator #14# like the other fractions:

#114/7 = (114*2)/(7*2) = 228/14#

So our express can be rewritten:

#228/14-40/14+9/14 = (228-40+9)/14#

Then note that addition and subtraction have the same priority, so we need to evaluate them from left to right:

#(color(blue)(228-40)+9)/14 = (188+9)/14 = 197/14#

To express this as a mixed number, we divide #197# by #14# to give a quotient #14# and remainder #1#, so:

#197/14 = 14+1/14 = 14 1/14#

1

Answer:

#20st = 2xx2xx5xxsxxt#

Explanation:

You need to write the given term or expression as the product of its prime factors. This is sometimes also called expanded form.

#20st = 2xx2xx5xxsxxt" "larr# these are all prime factors.

The variables are really easy to do because even if there are indices, you can see exactly how many of each there are.

So #36s^3t^2# would be expanded as:

#2xx2xx3xx3xxsxxsxxsxxtxxt#

1

Answer:

Change them to decimals and show each in the correct position on the number line.

Explanation:

Before you can compare any numbers, they have to be in the same form. Here you have decimals and fractions. Convert everything to decimals which are easier to compare than fractions.

The negative values are obviously smaller than the positives.

#-3/8 = -0.125#

#-0.3 = -0.300" "larr# the smallest value

#0.27 =0.270#

#5/8 = 0.625" "larr# the biggest value

On a number line, the numbers in order from the left would be:

#-0.3" "-0.125" "0.27" "0.625#

Make sure that the number line is marked with a suitable scale and use a dot to show exactly where each number lies.

2

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.

1

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.

1

Answer:

Conventionally, left to right but it doesn't matter.

Explanation:

We have a lot of freedom with this problem. Just do one thing at a time, and you should get it.

#-3-9+6+12-5#
#-12+6+12-5#
#-6+12-5#
#6-5#
#1#

We might ask ourselves though. Can we do it in any other order?

Let's add first.

#-3-9+6+12-5#
#-3-9+18-5#
#-12+18-5#
#6-5#
#1#

What about if we turn it into a purely addition problem?

#(-3)+(-9)+6+12+(-5)#
#(-3)+(-9)+18+(-5)#
#(-12)+18+(-5)#
#(-12)+13#
#1#

What about if we go from last to first?

#-3-9+6+12-5#
#-3-9+6+7#
#-3-9+13#
#-3+4#
#1#

The point is that you just do it. You can't go wrong. Addition and subtraction have the same precedence.

1

Answer:

there are an infinite amount of answers but here are some:

#13/60#

#7/30#

#15/60#

#4/15#

Explanation:

first let's find a common denominator just to make the problem easier.

#1/5# = #3/15# = .2

#1/3# = #5/15# = .333333333333

so essentially we want to find numbers larger than #3/15# and smaller than #5/15#.. . so #4/15# would be a great place to start. actually #4/15# could actually be one of your answers.

but let's have some fun with these numbers.

#3/15# = #6/30# = #12/60# we get this from multiplying the numerator and denominator by the same number (2).

#5/15# = #10/30# = #20/60# we get this from multiplying the numerator and denominator by the same number (2).

So to rephrase the question:
find 4 fractions between #12/60# and #20/60#

Here they are:

#13/60#

#14/60# this is the same as #7/30#

#15/60#

#16/60# this is the same as the #4/15# we saw earlier

also. . .

#17/60#

#18/60#

#19/60#

we can check by converting to decimal and seeing if the number is larger than .2 and smaller than .3333

when i put 13/60 in my calculator it says .21666 so I know it's right because
.2 < .216 < .333

1

Answer:

Get data into same units and compute

Explanation:

We can see that we have mixed units where some values are inches, and some are feet.

To properly reason about the problem, we must get everything in the same units.

We might be tempted to turn everything into feet, but since a foot is bigger than an inch, we will have fractional answers. That might make it more difficult, so it will be easier to convert feet to inches.

To do this, we must recall that there are 12 inches in 1 foot.

If it snowed 21.5 feet, and each foot has 12 inches, then it snowed #21.5*12=258# inches.

OK, well that might be reasonable. Let's look at the rest of our data.

Monday was 15 inches.
Tuesday was 4.5 inches.
Wednesday was 6.75 inches.

Just by looking at these values, we can see that we won't get anywhere near 258 inches, but let's be thorough.

#15+4.5+6.75=26.26# inches.

We now know the weather reporter lied to us, and since we already converted the units, we can find out by how much.

#258-26.26=231.74# inches

How many feet? Well, we just reverse the process. We can divide by 12 inches to get the total feet.

#231.74/12=19.311# feet.

But realistically, I think this problem might be one of those glance-and-solve types.

You quickly add up the inches to get 26.26 inches, and you already know that 21.5 feet is a great deal bigger than that, so you immediately answer no, it's not valid.