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In this video, we will look at comparing ratios using unit rates in real-life problems.
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We will begin by recapping how we can simplify ratios.
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If we consider the ratios four to 10 and six to 18, we can simplify them by looking for common factors.
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Four and 10 have a highest common factor of two, so we can divide both sides of the ratio by two.
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This tells us that the ratio four to 10 in its simplest form is two to five.
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We can repeat this process for the ratio six to 18.
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The highest common factor this time is six.
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As six divided by six is equal to one and 18 divided by six is equal to three, the ratio in its simplest form is one to three.
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This is all well and good if we want a ratio in its simplest form.
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But what if we wanted to compare the two ratios?
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How can we compare two to five and one to three?
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In order to compare two or more ratios, we need to write them in the form one to đť‘›.
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This is known as the unit ratio or unit rate.
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In terms of our example, the second ratio, one to three, is already written in this form.
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This means that, for every one unit of the first part, we get three units of the second part.
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In order to write the ratio two to five as a unit ratio, we need to divide both sides by two.
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Two divided by two is equal to one.
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Five divided by two can be written as the fraction five-halves or five over two.
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When comparing ratios, it is useful to turn this fraction into a decimal.
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The ratio four to 10 or two to five written as a unit ratio is one to 2.5.
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For every one unit of the first part, we get two and a half or 2.5 units of the second part.
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Weâ€™re now in a position to compare the ratios as required in the question.
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An alternative method to compare ratios is to consider the first part as a fraction of the whole.
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In our ratio four to 10, the first part four is four out of 14 parts altogether.
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In the same way, the first part of the second ratio, six to 18, is six parts out of 24 in total.
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Both of these fractions can be simplified by dividing the numerator and denominator by two and six, respectively.
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We still have a problem when trying to compare two-sevenths and one-quarter.
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The easiest way to do so would be to find the lowest common multiple of four and seven.
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This is 28.
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Two-sevenths is equivalent to eight twenty-eighths, whereas one-quarter is equivalent to seven twenty-eighths.
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As the denominators are now equal, we can compare the fractions by looking at the numerators.
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Whilst the second method is sometimes useful, for the majority of this video, we will compare ratios using unit ratios or unit rates.
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Scarlett uses two tablespoons of sugar for every three glasses of lemonade, while Natalie uses three tablespoons of sugar for every six glasses of lemonade.
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Who makes the sweeter lemonade?
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We can begin this question by writing down a ratio of sugar to lemonade for both of the girls.
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Scarlett used two tablespoons for every three glasses of lemonade.
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Therefore, her ratio is two to three.
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Natalie used three tablespoons of sugar for every six glasses of lemonade.
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So, her ratio is three to six.
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One way of comparing two or more ratios is to write them in the form one to đť‘›.
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This is known as the unit ratio.
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When finding an equivalent ratio, we must divide or multiply both sides by the same value.
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Two divided by two is equal to one.
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And three divided by two is equal to 1.5.
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This means that, for every one tablespoon of sugar that Scarlett uses, she will fill 1.5 glasses of lemonade.
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Repeating this process for Natalie, we divide both sides of her ratio by three.
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The ratio three to six simplifies to one to two.
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This means that, for every one tablespoon of sugar that Natalie uses, she is able to fill two glasses of lemonade.
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Weâ€™re asked to work out who makes the sweeter lemonade.
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This will be the person who has less glasses per tablespoon of sugar.
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As Scarlett is only able to make one and a half glasses of lemonade for one tablespoon of sugar, she makes the sweeter lemonade.
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Had we noticed that Natalie had double the number of glasses originally than Scarlett, we couldâ€™ve used an alternative method for this question.
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Multiplying both sides of Scarlettâ€™s ratio by two gives us a new ratio, four to six.
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As Scarlett used four tablespoons of sugar for six glasses of lemonade whereas Natalie only used three tablespoons, once again, we have proven that Scarlett made the sweeter lemonade.
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Michael wants to sign up for his schoolâ€™s sports competition.
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In order to be accepted, he has to be able to run 400 meters in one minute.
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Michael took 20 seconds to run 100 meters.
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If he could run at the same rate, would he qualify to take part?
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Weâ€™re told in the question that Michael needs to be able to run 400 meters in one minute.
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If we write this as a ratio of time in minutes to distance in meters, this would be one to 400.
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Weâ€™re also told that Michael can run 100 meters in 20 seconds.
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There are 60 seconds in one minute.
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And 20 multiplied by three is equal to 60.
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Multiplying 100 by three gives us 300.
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So, if Michael runs at the same rate, he will cover 300 meters in 60 seconds.
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This ratio of time in minutes to distance in meters is one to 300.
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As Michael needed to cover a distance of 400 meters in one minute, the correct answer is no.
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He would not qualify to take part.
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Writing any ratio in this form, one to đť‘›, is known as the unit ratio or unit rate.
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For every one unit, or minute in this case, of time, we can see the distance in meters that Michael covered.
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The next question we look at, weâ€™ll look at unit rates to compare three different ratios.
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Mason, Liam, and James are biking.
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Mason can bike two miles in 20 minutes, Liam can bike three miles in 25 minutes, and James can bike six miles in 66 minutes.
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Who cycles at the fastest rate?
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In order to compare the three speeds, we will write the ratio of distance in miles to time in minutes.
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For Mason, this is a ratio of two to 20.
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For Liam, the ratio is three to 25.
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And finally, for James, the ratio is six to 66.
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In order to compare the three ratios, we need to calculate the unit rate or unit ratio.
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This is written in the form one to đť‘›.
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In this question, this will calculate the time it would take each boy to cycle a distance of one mile.
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When simplifying or finding equivalent ratios, we need to multiply or divide both sides by the same number.
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For mason, we need to divide both sides by two.
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This means that the ratio two to 20 is equivalent to one to 10.
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It takes Mason 10 minutes to bike one mile.
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To make the left-hand side of Liamâ€™s ratio equal to one, we need to divide both sides by three.
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25 divided by three is equal to eight and one-third or 8.3 recurring, written with a dot or bar above the three.
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It takes Liam eight and a third minutes to cycle one mile.
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Dividing both sides of Jamesâ€™s ratio by six gives us the new ratio one to 11.
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It takes James 11 minutes to cycle one mile.
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As all three ratios are now written in terms of their unit rate, we can compare them.
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The person who cycles at the fastest rate will be the person who takes the least amount of time to cycle one mile.
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In this question, this is Liam.
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The next question involves comparing ratios using tables.
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Use the table to determine which runners ran at the same rate.
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In order to answer this question, we will look at the ratio of each runner of their time in hours to distance in miles.
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For Liam, this is two to 10.
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He jogged 10 miles in two hours.
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Jamesâ€™s ratio was three to 18 as he ran 18 miles in three hours.
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The corresponding ratios for David and Michael were four to 20 and three to 12, respectively.
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In order to compare two or more ratios, we need to write them in the form one to đť‘›.
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This is the unit rate or unit ratio.
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In this question, it will be the distance that each runner jogged in one hour.
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For Liam, we will divide both sides of the ratio by two.
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This means that Liam ran a distance of five miles per hour.
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We divide the two parts of Jamesâ€™s ratio by three.
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This gives us a ratio of one to six.
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So, James ran six miles in one hour.
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Repeating this for David and Michael tells us that David ran five miles per hour and Michael ran four miles per hour.
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Weâ€™ve been asked to identify which runners ran at the same rate.
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As both Liam and David had the same unit ratio of one to five, we can conclude that they ran at the same rate.
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Our final question will involve using a calculator to calculate density to compare populations.
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Daniel and Charlotte both have a keen interest in gardening and are concerned by the number of slugs that they keep finding in their respective vegetable patches.
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They want to compare the number of slugs in each of their gardens.
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But due to the differences in size of their vegetable patches, they decide to compare the number of slugs per square foot.
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Danielâ€™s vegetable patch is a rectangle with dimensions five foot by three foot.
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And Charlotteâ€™s is a circular patch with a radius of three foot.
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One Saturday morning, Daniel counts 21 slugs in his entire vegetable patch and Charlotte counts 36.
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There are three parts to this question.
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Work out the density of slugs in Danielâ€™s vegetable patch.
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Work out the density of slugs in Charlotteâ€™s vegetable patch.
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Who has the more severe slug problem?
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Weâ€™re told that Danielâ€™s patch is rectangular and measures five foot by three foot, whereas Charlotteâ€™s is circular with a radius of three foot.
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There were 21 slugs in Danielâ€™s vegetable patch and 36 in Charlotteâ€™s.
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We will now clear some space to calculate the number of slugs they had per square foot.
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Letâ€™s consider Danielâ€™s vegetable patch first.
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His patch was rectangular with dimensions three foot and five foot, and he found 21 slugs in his patch.
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We can calculate the area of any rectangle by multiplying the length by the width.
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In this case, we need to multiply five by three.
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This is equal to 15.
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Therefore, Danielâ€™s patch has an area of 15 square foot.
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In order to calculate the density per square foot, we can, firstly, write the ratio of the area to the number of slugs.
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This is equal to 15 to 21.
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To calculate the density of slugs in Danielâ€™s patch, we need to calculate the unit ratio, how many slugs there are per square foot.
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This is written in the form one to đť‘›.
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We divide both sides of the ratio by 15, giving us the ratio one to 1.4.
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The density of slugs in Danielâ€™s vegetable patch is therefore 1.4 slugs per square foot.
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We can now repeat this process for Charlotte.
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Charlotteâ€™s vegetable patch was circular and had a ratio [radius] of three foot.
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She found 36 slugs in her vegetable patch.
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The area of any circle can be calculated by multiplying đťś‹ by the radius squared.
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In this question, this is equal to đťś‹ multiplied by three squared.
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This is equal to 28.2743 and so on.
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This means that the area of Charlotteâ€™s vegetable patch is 28.27 square feet.
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For the purposes of this question, we will keep this as nine đťś‹.
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The ratio of area to slugs for Charlotte is therefore nine đťś‹ to 36.
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To find the unit ratio or density, we can divide both sides by nine đťś‹.
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36 divided by nine đťś‹ is equal to 1.273 and so on.
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Rounding this to one decimal place gives us 1.3 slugs per square foot.
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The three correct answers are 1.4, 1.3, and Daniel.
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As 1.4 is greater than 1.3, Daniel has the more severe slug problem.
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We will finish this video by summarizing the key points.
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In order to compare two or more ratios, we need to calculate the unit rate or unit ratio.
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This is written in the form one to đť‘›.
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To simplify a ratio or find an equivalent ratio, we must multiply or divide all parts of the ratio by the same number.
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For example, the ratio four to 12 can be written as a unit ratio by dividing both parts by four.
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The ratio four to 12 is equivalent to the unit ratio one to three.
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For every one unit of the first part, we have three units of the second part.