radio

Divide 1 by the bottom number of a fraction and then multiply that by the top number.

The price the tv sold at was 95% of the original price.

When you decrease by a percentage, you do number times 1-percentage as a decimal. In this case, the original price was multiplied by 0.95 to get £617.50.

To reverse this, divide £617.50 by 0.95 to get the original price.

$${\frac{{\mathtt{617.5}}}{{\mathtt{0.95}}}} = {\mathtt{650}}$$

The original price was 650.

Note, this is not the same as multiplying by 1.05 because if you do that, you find 5% of the reduced price added onto the reduced price. By dividing by 0.95, we get back to the original number.

Get all the x terms on one side and then get x on its own.

In the example you gave, we have 5x + 7x = 60.

To get x on its own, we factorise so

x(5+7)=60

Now x is on its own and we have $${\frac{{\mathtt{60}}}{\left({\mathtt{5}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}\right)}} = {\mathtt{5}}$$

x=5

440 people = 100%

108 people = x%

We can divide 440 by 440 to get one person. Divide 100 by 440 to get 1 person as a percentage.

$${\frac{{\mathtt{440}}}{{\mathtt{440}}}} = {\mathtt{1}}$$

$${\frac{{\mathtt{100}}}{{\mathtt{440}}}} = {\frac{{\mathtt{5}}}{{\mathtt{22}}}} = {\mathtt{0.227\: \!272\: \!727\: \!272\: \!727\: \!3}}$$

Then multiply by 108 to get 108 people as a percentage

$${\mathtt{0.227\: \!272\: \!727\: \!272\: \!7}}{\mathtt{\,\times\,}}{\mathtt{108}} = {\frac{{\mathtt{270}}}{{\mathtt{11}}}} = {\mathtt{24.545\: \!454\: \!545\: \!451\: \!6}}$$

You could also divide 440 by 108 and use the answer of that to divide by 100.

$${\frac{{\mathtt{440}}}{{\mathtt{108}}}} = {\frac{{\mathtt{110}}}{{\mathtt{27}}}} = {\mathtt{4.074\: \!074\: \!074\: \!074\: \!074\: \!1}}$$

$${\frac{{\mathtt{100}}}{{\mathtt{4.074\: \!074\: \!074\: \!074\: \!074\: \!1}}}} = {\frac{{\mathtt{270}}}{{\mathtt{11}}}} = {\mathtt{24.545\: \!454\: \!545\: \!454\: \!545\: \!5}}$$

$${\frac{{\left({\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}^{{\mathtt{2}}}}{{\mathtt{3}}}} = {\mathtt{25}}$$

Times both sides by 3

$${\left({\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}^{{\mathtt{2}}} = {\mathtt{75}}$$

Square root both sides

$${\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}} = {\sqrt{{\mathtt{75}}}}$$

$${\mathtt{x}} = {\sqrt{{\mathtt{75}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}$$

$${\mathtt{x}} = {\sqrt{{\mathtt{75}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}} \Rightarrow {\mathtt{x}} = {\mathtt{9.660\: \!254\: \!037\: \!844\: \!386\: \!5}}$$

Remeber SOH CAH TOA.

You have the oppoisite and hypotenuse whic are to O and H, so we use sine or sin().

Because we don't know the angle, we need to rearrange the equation to get the angle on its own. In this case, the angle will be x and O and H are known numbers.

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{x}}\right)} = {\frac{{\mathtt{O}}}{{\mathtt{H}}}}$$

$${\mathtt{x}} = \underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{O}}}{{\mathtt{H}}}}\right)}$$

Note I didn't use sin(O/H), i used sin^-1(O/H), which is the inverse of sine.

We can reverse the question so we start with -36.

$${\mathtt{\,-\,}}{\mathtt{36}}{\mathtt{\,-\,}}{\mathtt{7}} = -{\mathtt{43}}$$

$${\mathtt{\,-\,}}{\frac{{\mathtt{43}}}{{\mathtt{4}}}} = -{\mathtt{10.75}}$$

So the number is -10.75

We can check by doing it forwards again.

$${\mathtt{\,-\,}}\left({\mathtt{10.75}}{\mathtt{\,\times\,}}{\mathtt{4}}\right) = -{\mathtt{43}}$$

$${\mathtt{\,-\,}}{\mathtt{43}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}} = -{\mathtt{36}}$$

Ahh. Forgot to combine on each step.

We can find out how far each travels in one second by dividing the distance by the time taken.

$${\frac{{\mathtt{150}}}{{\mathtt{40}}}} = {\frac{{\mathtt{15}}}{{\mathtt{4}}}} = {\mathtt{3.75}}$$

$${\frac{{\mathtt{40}}}{{\mathtt{16}}}} = {\frac{{\mathtt{5}}}{{\mathtt{2}}}} = {\mathtt{2.5}}$$

150 yards in 40 seconds is faster because in one second, you travel 3.75 yards, but 40 yards in 16 seconds is only 2.5 yards a second.

Linear inequalities are very similar to linear equations.

Here is an example of a linear inequality and equation.

x+3=0

x+3<0

Move x to one side and the numbers to the other side while grouping like term.

x=3

x<3

When you multiply or divide by a negative number in inequalities, you need to remember to flip the inequality sign around.

-x<3

x>-3

or

x/-2>12

x<-24