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In this video, we will learn how to simplify and use the ratio between three numbers to solve problems.
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We will begin by recalling what we mean by ratio.
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A ratio shows the relative sizes of two or more values.
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In this video, we will look at questions with three values.
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Ratios can be shown in different ways, firstly, using a colon to separate example values, secondly, written as fractions to separate one value from the total.
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Any fraction can be converted to a decimal by dividing the numerator by the denominator, in this case, dividing one value by the total.
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Finally, we could convert a decimal to a percentage by multiplying the decimal by 100.
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As an example, letβs imagine we had one boy and three girls.
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This can be written as a ratio one to three, where the first number represents the boys and the second the girls.
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When dealing with ratio, it is important the order is correct.
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If we wanted the ratio of girls to boys, this would be three to one.
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As there are four children in total and one is a boy, one-quarter of the children are boys and three-quarters are girls.
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These can be converted to decimals by dividing one by four and three by four, giving us 0.25 and 0.75.
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This means that 25 percent of the children are boys and 75 percent are girls.
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The first question that weβll look at involves simplifying a ratio.
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Express the ratio 72 : 54 : 81 in its simplest form.
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In order to simplify any ratio, we must divide each of the numbers by the same value.
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This means that weβre looking for a common factor of 72, 54, and 81.
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Whilst it will be quicker if we choose the highest common factor, any common factor will enable us to start the question.
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72, 54, and 81 are all in the nine times table.
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Therefore, theyβre all divisible by nine.
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72 divided by nine is eight, 54 divided by nine is six, and 81 divided by nine is equal to nine.
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This means that the ratio simplifies to eight to six to nine.
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These three numbers have no common factor except one.
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This means that the ratio is in its simplest form.
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Whilst eight and six are exactly divisible by two, nine is not.
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Likewise, six and nine are divisible by three, but eight is not.
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The ratio 72 : 54 : 81 written in its simplest form is eight : six : nine.
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Our next question involves combining two ratios to find the ratio between three numbers.
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Determine the ratio between the three numbers π, π, and π given that the ratio of π to π is 10 to one and the ratio of π to π is two to one.
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Weβre asked to work out the ratio of π to π to π.
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Weβre told in the question that the ratio of π to π is 10 to one.
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Weβre also told that the ratio of π to π is two to one.
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π occurs in both of these.
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So we need to use equivalent ratios to make sure the value of π is the same.
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We can find equivalent ratios by multiplying all of our values by the same number.
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This means that the ratio 20 to two is equivalent to the ratio 10 to one as we have multiplied both values by two.
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The ratio of π to π can therefore be rewritten as 20 to two.
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As the value for π is now the same, the ratio of π to π to π is 20 : two : one.
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Our next two questions are ratio problems in context.
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The ratio between the heights of three buildings A, B, and C is 10 : four : three.
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If the height of building A is 60 meters, find the heights of building B and building C.
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Weβre told in the question that the ratio of the heights is 10 : four : three.
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Weβre also told that the height of building A is 60 meters.
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We can find equivalent ratios by multiplying each of our values by the same number.
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10 multiplied by six is equal to 60.
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This means that, in order to find an equivalent ratio, we must also multiply four and three by six.
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Four multiplied by six is 24, and three multiplied by six is 18.
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This means that the ratio 10 : four : three is equivalent to the ratio 60 : 24 : 18.
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We can therefore conclude that the height of building B is 24 meters and the height of building C is 18 meters.
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The total number of students in the first, second, and third grades at a primary school is 285 in the ratio seven : four : eight.
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Calculate the number of students in each grade.
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We can share a total, in this case, 285 students, in a given ratio by following three steps.
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Firstly, we find the sum of the ratios.
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Seven plus four plus eight is equal to 19.
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Our second step is to divide the total by this answer.
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285 divided by 19 is equal to 15.
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This is equal to one part or one share of the ratio.
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Our final step is to multiply the value for one part by each of the ratios.
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15 multiplied by seven is 105.
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15 multiplied by four is equal to 60.
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15 multiplied by eight is equal to 120.
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As the seven corresponded to the number of first graders, there are 105 students in first grade, 60 students in second grade, and 120 students in third grade.
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It is always worth checking our answers by adding the values to ensure this makes the total.
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105 plus 60 plus 120 is equal to 285.
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This method works for sharing any total between any number of ratios.
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The last two questions we will look at are more complicated problems involving algebra.
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If 10π₯ is equal to 11π¦ which is equal to 12π§, find the ratio of π₯ to π¦ to π§.
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There are lots of ways of solving this problem.
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One way would be to find values that solve different parts of the equation first.
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Letβs begin by considering 10π₯ is equal to 11π¦.
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Substituting in the values π₯ equals 11 and π¦ equals 10 would mean that this equation is true.
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10 multiplied by 11 and 11 multiplied by 10 are both equal to 110.
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This means that the ratio of π₯ to π¦ could be written as 11 to 10.
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Letβs now consider the fact that 10π₯ is also equal to 12π§.
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In this equation, π₯ equals 12 and π§ equals 10 is a solution.
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10 multiplied by 12 and 12 multiplied by 10 are both equal to 120.
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This means that the ratio of π₯ to π§ is 12 to 10.
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We now have two ratios, a ratio of π₯ to π¦ and a ratio of π₯ to π§.
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In order to combine these ratios, we need to use equivalent ratios to ensure that the value for π₯ is the same.
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The lowest common multiple of 11 and 12 is 132.
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We can therefore multiply the top ratio by 12 and the bottom ratio by 11.
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The ratio 11 to 10 is equivalent to the ratio 132 to 120.
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Likewise, the ratio of π₯ to π§ of 12 to 10 is equivalent to 132 to 110.
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As our value for π₯ is the same, we can now combine the ratios.
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The ratio of π₯ to π¦ to π§ is 132 : 120 : 110.
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This ratio can be simplified as all of our values are even and are therefore divisible by two.
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132 divided by two is equal to 66.
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120 divided by two is equal to 60.
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And 110 divided by two is equal to 55.
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The ratio of π₯ to π¦ to π§ in its simplest form is 66 : 60 : 55 as these three numbers have no common factor apart from one.
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If the ratio of π₯ to π¦ to π§ is three : four : eight, then what is the value of π₯ squared plus π¦ squared plus π§ squared divided by π¦ multiplied by π₯ plus π§.
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We can see from the ratio given that our value of π₯ is equal to three, our value of π¦ is equal to four, and our value of π§ is equal to eight.
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We can substitute these values directly into our expression.
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Three squared is equal to nine, four squared is 16, and eight squared is 64.
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So the numerator simplifies to nine plus 16 plus 64.
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We can simplify the denominator by using our order of operations known as PEMDAS or BIDMAS.
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We do the parentheses or brackets first, leaving us with four multiplied by 11.
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Nine plus 16 plus 64 is equal to 89.
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Four multiplied by 11 is 44.
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The value of the expression π₯ squared plus π¦ squared plus π§ squared divided by π¦ multiplied by π₯ plus π§ is 89 over 44.
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This fraction can not be simplified as 89 and 44 have no common factor apart from one.
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We will now summarize the key points from this video.
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We recalled at the start of the video that a ratio shows the relative sizes of two or more values.
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In this video, we were dealing with problems involving three values, where the ratios were written in the form π₯ to π¦ to π§.
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The questions that we answered included simplifying ratios, dividing a quantity using a ratio, and algebraic problems.
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Important topics that help us solve problems involving ratio include factors, multiples, and basic arithmetic.