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In this video, we will learn how to multiply rational numbers including fractions, decimals, and percentages.
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We will begin by recalling what we mean by a rational number and how we can convert between decimals, fractions, and percentages.
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A rational number is a number that can be written as a fraction in the form ๐ over ๐ where ๐ and ๐ are integers and ๐ is not equal to zero.
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This means that all whole numbers or integers are rational numbers.
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It doesnโt matter if theyโre positive or negative, as nine can be written as nine over one and negative seven as negative seven over one.
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It follows that all positive and negative fractions are rational numbers as they are written in the form ๐ over ๐.
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Any mixed number is also a rational number.
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For example, two and a half can be rewritten as the improper fraction five over two.
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Decimals of the form 0.2 and 4.31 are also rational, likewise any percentage, for example, 25 percent.
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0.2 is equal to two-tenths.
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This can be simplified to one-fifth by dividing the numerator and denominator by two.
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It is also equal to 20 percent.
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We can convert any decimal into a percentage by multiplying by 100.
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In this video, we will need to convert between decimals, fractions, and percentages to find the most appropriate value.
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We do this in order to make our calculation as simple as possible.
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Our first question involves multiplying two fractions.
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Evaluate negative seven-fifths multiplied by three-quarters.
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We recall that when multiplying two fractions, we simply multiply the numerators and separately multiply the denominators.
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Where possible, we can cross cancel or cross simplify first.
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We also recall that multiplying a negative number by a positive number gives us a negative answer.
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In this question, weโre multiplying negative seven-fifths by positive three-quarters.
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This means that our answer must be negative.
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Multiplying the numerators gives us 21 as seven multiplied by three is 21.
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Five multiplied by four is equal to 20, so the denominator equals 20.
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Negative seven-fifths multiplied by three-quarters is equal to negative twenty-one twentieths or negative 21 over 20.
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We could convert this into a mixed number by dividing 21 by 20.
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This is equal to one remainder one.
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Therefore, negative 21 over 20 is the same as negative one and one twentieths.
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Our next question involves multiplying a mixed number by a negative fraction.
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Calculate two and three-fifths multiplied by negative two-sevenths.
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Give your answer as a fraction in its simplest form.
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Our first step in this question is to convert the mixed number two and three-fifths to a top heavy or improper fraction.
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In the bar shown, we have shaded two and three-fifths.
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Each complete bar is equal to five-fifths.
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This means that altogether, we have thirteen-fifths shaded.
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The fraction two and three-fifths is equal to thirteen-fifths.
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A quicker way of calculating this is to multiply the whole number by the denominator and then adding the numerator.
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Two multiplied by five is 10, and adding three gives us 13.
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This is the numerator of our improper fraction.
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We, therefore, need to multiply thirteen-fifths by negative two-sevenths.
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We recall that when multiplying two fractions ๐ over ๐ and ๐ over ๐, we simply multiply the numerators and separately multiply the denominators.
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We also need to remember that when we multiply a positive number by a negative number, we get a negative answer.
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13 multiplied by two is 26.
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Five multiplied by seven is 35.
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Multiplying positive thirteen-fifths by negative two-sevenths is equal to negative 26 over 35 or negative twenty-six thirty-fifths.
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Our next question is a worded problem in context.
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Sara works in a supermarket.
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She earns seven dollars per hour.
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How much will she get paid if she puts in 35 and one-quarter hours per week?
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Write your answer as a decimal.
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Saraโs pay will be equal to her wage per hour multiplied by the number of hours she works.
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We are told she earns seven dollars per hour.
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We are also told that she works for 35 and a quarter hours per week.
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We need to multiply this by seven.
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There are lots of ways of working out this calculation.
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One way would be to multiply seven by 30, seven by five, and seven by a quarter.
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As seven multiplied by three is 21, seven multiplied by 30 is 210.
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Seven multiplied by five is 35.
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Seven multiplied by one-quarter is seven-quarters.
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As seven divided by four is equal to one remainder three, seven-quarters is the same as the mixed number one and three-quarters.
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We know that three-quarters is equal to the decimal 0.75.
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Therefore, one and three-quarters is equal to 1.75.
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We need to add 210, 35, and 1.75.
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This is equal to 246.75.
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If Sara earns seven dollars per hour and works for 35 and a quarter hours, she will earn 246 dollars and 75 cents.
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In our next question, we will multiply a percentage by a mixed number.
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Calculate 50 percent of one and a half.
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Give your answer as a fraction.
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In order to answer this question, we firstly need to convert 50 percent into a fraction.
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As the word โpercentโ means out of 100, we can convert from a percentage to a decimal by dividing by 100.
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50 divided by 100 is 0.5.
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This is equal to five-tenths.
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We can then simplify this fraction by dividing the numerator and denominator by five.
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50 percent is equal to one-half.
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Next, we need to convert 1.5 into a top heavy or improper fraction.
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There are two-halves in one whole one.
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Therefore, one and a half is equal to three-halves or three over two.
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We know that the word โofโ in mathematics means multiply.
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We need to multiply one-half by three-halves.
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When multiplying two fractions, we multiply the two numerators and then the two denominators separately.
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One multiplied by three is three, and two multiplied by two is four.
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We can, therefore, conclude that 50 percent of one and a half is three-quarters.
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We could also have shown this pictorially.
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We began with one and a half and wanted to calculate 50 percent or a half of this.
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One-half of a whole one is equal to one-half, and one-half or 50 percent of a half is a quarter.
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Adding a half and a quarter once again gives us a final answer of three-quarters.
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The penultimate question in this video involves multiplying a percentage by a decimal.
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Calculate 25 percent multiplied by 0.2.
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Give your answer as a decimal number.
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As we need to give our answer as a decimal number, we firstly need to convert 25 percent into either a decimal or a fraction.
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The word โpercentโ means out of 100.
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Therefore, 25 percent is equal to 25 over or out of 100.
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As the line in the fraction means divide, this is equal to 0.25.
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The fraction can also be simplified by dividing the numerator and denominator by 25.
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25 percent is, therefore, equal to one-quarter.
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This means that we have two options to proceed.
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We can either multiply one-quarter by 0.2 or 0.25 by 0.2.
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Multiplying by a quarter is the same as dividing by four.
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We need to divide 0.2 by four.
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This is equal to 0.05.
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When multiplying the two decimals 0.25 and 0.2, we know that 25 multiplied by two is 50.
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As there are three digits altogether after the decimal point in the question, there needs to be three digits after the decimal point in the answer.
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0.25 multiplied by 0.2 is equal to 0.050.
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This is the same as 0.05.
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25 percent multiplied by 0.2 written as a decimal is 0.05.
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Our final question involves multiplying an integer, a fraction, and a decimal.
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Calculate 25 multiplied by one-sixth multiplied by 0.08.
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Give your answer as a fraction in its simplest form.
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As we need to give our answer as a fraction, our first step is to convert 0.08 into a fraction.
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As the eight is in the hundredths column, this is equal to eight hundredths or eight over 100.
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Both the numerator and denominator are divisible by four, so this fraction simplifies to two over 25 or two twenty-fifths.
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We need to multiply 25, one-sixth, and two twenty-fifths.
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Any integer or whole number, in this case, 25, can be written over one.
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Before multiplying the fractions, we can now cross simplify or cross cancel.
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There is a 25 on the numerator and denominator, so these will cancel.
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The numbers two and six are both even, so they are divisible by two.
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On the numerators, we are left with one multiplied by one multiplied by one.
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And on the denominator, we are left with one multiplied by three multiplied by one.
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25 multiplied by one-sixth multiplied by 0.08 is, therefore, equal to one-third.
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We will now summarize the key points from this video.
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A rational number is any number that can be written as a fraction where the numerator and denominator are integers and the denominator cannot be equal to zero.
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Rational numbers include positive and negative integers, fractions, and any recurring or terminating decimal.
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They also include percentages.
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Where there is a mixture of these, we need to convert them all into either fractions or decimals.
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When multiplying two or more fractions, we multiply the numerators and denominators separately.
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However, it is useful to cross cancel or cross simplify first.
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Finally, we need to remember that when we multiply two positive or two negative numbers, we get a positive answer, whereas when we multiply a positive by a negative in either order, we get a negative answer.