Lesson video

Hi, everyone.

<v ->My name is Ms. Coe</v> and today I'm really excited to be learning with you as we'll be looking at the wonderful unit ratio.

I hope you enjoy the lesson.

So let's make a start.

Hi, everyone and welcome to this lesson on algebraic ratios under the unit ratio.

And by the end of the lesson, you'll be able to form equations from algebraic ratios and solve to find an unknown.

The keywords are variables and variables are in proportion if they have a constant multiplicative relationship.

We'll also be looking at ratio and a ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

Finally, we'll be looking at factorising and to factorise is to express a term as the product of its factors.

Today's lesson will be broken into two parts.

We'll be looking at interpreting algebraic ratios first and then forming and solving equations.

So let's make a start interpreting algebraic ratios.

Now, equivalent ratios are equivalent because the proportions between parts is the same.

For example, we know two to three is equivalent to four to six.

This is because the ratio between the parts is the same and we can see this by two over the three is exactly the same as four over the six or the ratio of these parts, two over the four is exactly the same as three over the six.

Or we can see it using these parts.

Four over the two is exactly the same as six over the three.

But can you spot anymore? There is one more there.

Let's see if you can spot it.

Well done.

Well, it should be 3/2 is exactly the same as 6/4.

So this can also be seen putting our ratio into a ratio table.

So let's insert the ratio of two to three and four to six into the ratio table.

And you can see it here.

You can see that multiplicative relationship again, 3/2 and 6/4.

These are the same because of that multiplier of 3/2.

Same again, it can also be seen here, two over the three is exactly the same as 4/6 and that's because of this multiplicative relationship here.

Alternatively, you can see it here, 4/2 is the same as 6/3 because of this multiplicative relationship.

And lastly, you can see here two over the four is exactly the same as three over the six because of this multiplicative relationship.

So the proportional relationship between the parts of the equivalent ratios can easily be seen using a ratio table.

So therefore, understanding these proportional relationships allows you to write an algebraic ratio as an equation.

For example, we're asked to form a single equation where we know x minus one to x plus one is equal to 2x plus one to 5x.

Now, we can put this in a ratio table.

Then we can identify those equivalent ratios.

Now remember, we're only asked to form a single equation, but there are quite a few here to choose from.

I'm going to choose x subtract one over the 2x plus one is equivalent to x plus one over our 5x.

So here, this is one equation formed using our knowledge of those proportional relationships.

Now, we can form another one, 2x plus one over the x minus one is equal to our 5x over our x plus one.

Same again, this is another algebraic equation formed from our algebraic ratio because of those proportional relationships.

Now, can you find two more equations when we are given the same ratios? So you can give it a go and I've left this ratio table here to help if you need.

Well done.

Well, hopefully you've got this equation and you also have this one.

Really well done if you got this.

So it's not essential to draw a ratio table because the equation can be formed without it.

However, it does help us ensure that we're comparing the correct parts of our ratio.

What I want us to do is another check and I want you to identify which of the following are the correct equations for this algebraic ratio? 2x to x plus five is equal to 3x plus four to 5x take away two.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, here are our correct ratios.

As you can see, I've used a ratio table to help us identify this.

Really well done if you got this one right.

Great work, everybody.

So now it's time for your task.

What I want you to do is put a tick next to the correct equation for the algebraic ratio.

4x plus five to seven is equal to 2x take away one to 4x.

Take your time, press pause if you need.

Well done.

Let's move on to question two.

Question two wants you to put a tick next to the correct equation for the algebraic ratio x to seven plus x is equal to x take away one to 4x plus nine.

See if you can give it a go.

Press pause if you need more time.

Great work.

Let's have a look at question three.

Question three shows Sam and his working out and this is the working out for a question.

X to x minus three is equal to x plus two to 2x plus one, and he uses this ratio table here, and he comes up with the following algebra equation.

Explain Sam's mistake and write the correct equation.

See if you can give it a go.

Press pause if you need more time.

Well done.

So let's move on to question four.

The ratio of a plus b to a minus b is equivalent to n to one and we have to show that n is equal to a plus b over a minus b.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's go through these answers.

Well, for question one, it's only a and c.

Massive well done if you spotted this.

Drawing a ratio table can help.

Well done.

Let's move on to question two.

Question two, you should have had the answers of a and b.

Really well done.

Drawing a ratio table can help.

For question three, did you spot Sam's mistake and write the correct equation? Well, Sam has incorrectly written the ratio table.

The ratio table should have been written like this and then from here, we can form the correct equation, which is x over x minus three is equal to x plus two over 2x plus one.

Really well done.

Lastly, for question four, well, I'm going to write my ratios now.

So it's a plus b to a minus b is equal to n to one.

Putting it into a ratio table, you can identify one of these equations.

I've written a plus b over a minus b is equal to n over one, and you can do a little bit more work if needed, but you can clearly see that n is equal to a plus b over a minus b.

Well done.

Great work, everybody.

So now it's time for the last part of our lesson, which is forming and solving equations.

So once an equation has been formed from an algebraic ratio, we can solve.

For example, the question wants us to work out the value of x given the algebraic ratio five to 3x plus three is equal to x plus two to x squared plus x.

Firstly, let's form an equation.

For me, I'm going to form this one.

So I've chosen this equation five over 3x plus three is equal to x plus two over x squared plus x.

Now, the next step is to factorise.

So to simplify if possible, so you might notice how I factorised 3x plus three into three bracket x plus one and I factorised x square plus x into x bracket x plus one.

Now, once you've done this, you can multiply so the denominators are the same.

So once you have the same denominator, this allows you to equate the numerator.

In other words, I'm creating equivalent algebraic fractions.

So on the left-hand side, you can see I have the fraction 5x over 3x, bracket x plus one, and on the right-hand side, I have the equation three bracket x plus two over 3x bracket x plus one.

I've made equivalent algebraic fractions with the same denominator.

Because the denominators are now the same, I can simply equate those numerators.

This means 5x is equal to 3x plus six because I've expanded out those brackets, which means 2x is equal to six, which then gives me the answer to x to be three.

This is a great question and the process of factorising and making the denominator the same allows us to equate those numerators so to efficiently solve.

So let's have a look at a check question.

The check question gives us the ratio five to x plus three is equal to four to x and we're asked to find the value of x.

Take your time with this and press pause.

Well done.

So let's see how you got on.

Well, for me, I'm going to form this algebraic equation.

Now, from here, you might spot I've got different denominators.

So I'm going to write equivalent denominators by multiplying my left fraction by x over x and multiply my right fraction by x plus three over x plus three.

This makes a common denominator.

Because my denominators are common, that means I can equate my numerators, giving me 5x is equal to 4x plus 12.

Solving for x means x is 12.

Really well done if you got this.

So equating the numerators when the denominators are equal is an efficient method, even when more complex algebraic ratios are given.

For example, x minus five to two minus x is equal to 10 to x plus two.

And we're asked to find the value of x where x is greater than zero, same as before.

Let's form an equation from my algebraic ratio.

I have formed this equation.

Now, looking at our denominators, I need to make the denominators the same.

So I'm going to multiply my left algebraic fraction by x plus two over x plus two and my right algebraic fraction by two minus x over two minus x.

This will then give me this same denominator.

Given the fact that my denominators are the same, I can equate my numerator.

This means x subtract five multiplied by x plus two is exactly the same as 10 multiplied by two minus x.

Now, from here, I'm going to expand and then simplify.

Remember, equating quadratics to zero allows us to easily factorise and solve.

So from here, we can get our two solutions.

Now, x is equal to three or x is equal to negative 10.

But the question does state that all our x values have to be greater than zero.

So therefore, x equals three is the only solution.

Now what I want you to do is another check question.

Given that x plus two to five minus 2x is equal to three to x take away 10, I want you to find the value for x where x is greater than zero.

So you can give it a go.

Press pause if you need more time.

Well done.

So let's see how you got on.

Well, I've formed this equation here and from here, let's make our common denominators.

So I'm going to multiply the left algebraic fraction by x minus 10 over x minus 10, and the right algebraic fraction by five minus 2x over five minus 2x.

As you can see, we now have a common denominator.

So I can equate those numerators, giving me x plus two multiplied by x minus 10 is equal to three bracket five minus 2x.

Now, from here, I'm going to expand, giving me x squared take away 8x take away 20 is equal to 15 take away 6x.

Now, remember, equating our quadratic to zero allows us to solve more efficiently.

X squared take away 2x take away 35 is equal to zero, which can be factorised into x take away seven multiplied by x plus five.

This gives a zero, meaning our solutions can either be x is equal to seven, or x is equal to negative five.

Remember, the question told us x must be greater than zero.

So therefore, the only solution we have for x is seven.

Well done, great work, everybody.

So now it's time for your task.

I want you to work our the value of x given the following algebraic ratios.

And the question states when x is greater than zero.

For A, we have six to x minus three is equal to x plus 20 to 13.

See if you can give it a go.

Show all your working out.

Press pause as you'll need more time.

Well done.

For B, we have eight to x plus one equals x plus seven to 3x take away seven.

See if you can give it a go.

Show all your working out.

Press pause as you'll need more time.

Well done.

And for C, we have 2x to x plus five is equal to 3x plus four to 5x take away two.

See if you can give it a go.

Show all your working out.

Press pause as you'll need more time.

Great work.

So let's go through our answers.

For 1a, I've written this algebraic fraction here.

Remember, making those denominators the same means we can equate those numerators.

So I've formed my quadratic being x squared plus 17x take away 138 is equal to zero.

Factorising gives me two solutions.

That means x is equal to six or x is equal to negative 23.

Now, remember, the question states that x has got to be greater than zero, so therefore, x must be equal to six.

Press pause if you need to look at that working out a little bit more.

For B, we have this algebraic ratio, so this is my working out.

I have formed this algebraic fraction.

From here, I've made my denominators the same.

So I've equated those numerators.

Expanding and simplifying gives me the quadratic x squared minus 16x plus 63 is equal to zero.

Factorising means I do have two solutions.

I have x is equal to seven or x is equal to nine.

So I do have two solutions here.

Remember, the question, does want x to be greater than zero.

Press pause if you need to look at that working out a little bit more.

And for the last part, for C, we have this algebraic ratio and I've formed this algebraic fraction.

Making those denominators the same means I can equate those numerators, giving me a quadratic of 7x squared take away 23x take away 20 is equal to zero.

Factorising it gives me 7x plus five multiplied by x take away four is equal to zero.

Notice how I have a negative solution, which I'm not using.

So the only solution would be x is equal to four.

Press pause if you need to look at that working out a little bit more.

Great work, everybody.

So in summary, the proportional relationships between the parts of the equivalent ratios can be easily seen with and without a ratio table.

Therefore, understanding these proportional relationships allows you to write an algebraic ratio as an equation.

And it's important to remember, algebraic fractions follow the same rules as fractions and equating those numerators when the fractions have the same denominator is an efficient method.

And also remember, when a quadratic equation is formed, calculating two solutions is possible.

So it's important to reflect on the validity of each solution.

I hope you enjoyed this lesson, in particular, blending the work with ratio with the work of algebra.

And it's important to remember those processes that apply to numbers, such as fractions, also apply to algebra too.

Those processes don't change and that becomes really clear when using ratio tables too.

We can see that proportional relationship when using ratio tables, whether it be numerical or algebraic, and we can form those equivalent fractions or equivalent algebraic fractions really easily with those ratio tables.

I do hope you've enjoyed this lesson as much as I have enjoyed delivering it.

A huge well done, everybody.

It was great learning with you.