WEBVTT
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The height of tree A is π₯ centimeters.
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The height of tree B is π¦ centimeters.
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The height pf tree C is π§ over two centimeters.
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The height of tree B is 20 percent more than the height of tree A.
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The height of tree B is 40 percent less than the height of tree C.
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Work out the ratio π₯ to π¦ to π§.
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Write the ratio in its simplest form.
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To find a ratio of π₯ to π¦ to π§, we should begin by forming expressions for the height of the tree in one variable.
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Currently, we have three variables: they are π₯, π¦, and π§.
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We are to hold the relationship of the heights of trees A and C compared to B.
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So letβs form some equations for π₯ and π§ in terms of π¦.
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We are told that the height of tree B β remember that was π¦ centimeters β is 20 percent more than the height of tree A that was π₯ centimeters.
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Remember the original value is always 100 percent.
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So to increase by 20 percent, we can add 120.
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100 plus 20 is 120 percent.
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So an increase of 20 percent is the same as finding 120 percent of that same number.
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We can then find the decimal multiplier that corresponds to an increase of 20 percent by dividing this value by 100.
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When we divide by 100, we move the digits to the right two places.
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So 120 percent is equal to 1.2.
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To increase by 20 percent then, we multiply by 1.2.
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That means that π¦ is equal to 1.2 multiplied by π₯.
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Remember we wanted to form an equation for π₯ in terms of π¦.
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To achieve this, weβre going to divide both sides of this equation by 1.2.
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And in doing so, we get that π¦ over 1.2 is equal to π₯.
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In fact, if we type one divided by 1.2 into our calculator, we get five-sixths.
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So this tells us that five-sixths of π¦ is equal to π₯ or π₯ is equal to five-sixths of π¦.
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Letβs repeat this process with the information that the height of tree B is 40 percent less than the height of tree C.
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This time the height of tree B is π¦.
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But tree C is π§ over two.
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And of course, this time weβre reducing.
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So weβre going to subtract 40 percent from 100.
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Thatβs 60.
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So to find a 40 percent reduction, we find 60 percent of that number.
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Dividing by 100 gives us a decimal multiplier of 0.6.
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This time we can say that π¦ is equal to 0.6 lots of π§ over two.
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0.6 divided by two is 0.3.
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So π¦ is equal to 0.3π§.
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We want an expression for π§ in terms of π¦.
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So weβre going to divide both sides of this equation by 0.3.
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And we get π¦ divided by 0.3 is equal to π§.
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This time one divided by 0.3 is ten-thirds.
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And we can, therefore, see that π§ is equal to ten-thirds of π¦.
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Now that we have expressions for π₯ and π§ in terms of π¦, we can substitute all of these into the original ratio.
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π₯ is equal to five-sixths of π¦, π¦ is still π¦, and π§ was equal to ten-thirds of π¦.
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Remember weβre looking to write our ratio in its simplest form.
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And we can see that there is a common factor throughout of π¦.
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We can, therefore, divide everything through by π¦.
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We get five-sixths to one to ten-thirds.
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Finally, to write a ratio in its simplest form, we need these values to be integers β thatβs whole numbers.
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The quickest way to achieve this is to multiply through by the lowest common multiple of each of the denominators.
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The lowest common multiple of six and three is six.
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So weβre going to multiply each part of our ratio by six.
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Five-sixths multiplied by six is five, one multiplied by six is six, and ten-thirds multiplied by six is 20.
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Now, we are allowed to use a calculator here.
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But it is useful to know how to multiply a fraction by a whole number.
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Should it arise on a non-calculator paper, we give the whole number a denominator and its denominator is one.
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So ten-thirds multiplied by six is the same as ten-thirds multiplied by six over one.
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We could cross cancel or we could simply multiply the numerators together β 10 multiplied by six is 60 β and then the denominators β three multiplied by one is three.
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We can then see that 60 divided by three is 20, as we showed.
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In its simplest form then, the ratio of π₯ to π¦ to π§ is five to six to 20.