Lesson Video: Multiplying Rational Numbers | Nagwa Lesson Video: Multiplying Rational Numbers | Nagwa

Lesson Video: Multiplying Rational Numbers Mathematics • 7th Grade

In this video, we will learn how to multiply rational numbers, including fractions, decimals, and percentages.

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Video Transcript

In this video, we will learn how to multiply rational numbers including fractions, decimals, and percentages. We will begin by recalling what we mean by a rational number and how we can convert between decimals, fractions, and percentages.

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. This means that all whole numbers or integers are rational numbers. It doesnโt matter if theyโre positive or negative, as nine can be written as nine over one and negative seven as negative seven over one. It follows that all positive and negative fractions are rational numbers as they are written in the form ๐ over ๐.

Any mixed number is also a rational number. For example, two and a half can be rewritten as the improper fraction five over two. Decimals of the form 0.2 and 4.31 are also rational, likewise any percentage, for example, 25 percent. 0.2 is equal to two-tenths. This can be simplified to one-fifth by dividing the numerator and denominator by two. It is also equal to 20 percent. We can convert any decimal into a percentage by multiplying by 100.

In this video, we will need to convert between decimals, fractions, and percentages to find the most appropriate value. We do this in order to make our calculation as simple as possible. Our first question involves multiplying two fractions.

Evaluate negative seven-fifths multiplied by three-quarters.

We recall that when multiplying two fractions, we simply multiply the numerators and separately multiply the denominators. Where possible, we can cross cancel or cross simplify first. We also recall that multiplying a negative number by a positive number gives us a negative answer. In this question, weโre multiplying negative seven-fifths by positive three-quarters. This means that our answer must be negative. Multiplying the numerators gives us 21 as seven multiplied by three is 21. Five multiplied by four is equal to 20, so the denominator equals 20. Negative seven-fifths multiplied by three-quarters is equal to negative twenty-one twentieths or negative 21 over 20. We could convert this into a mixed number by dividing 21 by 20. This is equal to one remainder one. Therefore, negative 21 over 20 is the same as negative one and one twentieths.

Our next question involves multiplying a mixed number by a negative fraction.

Calculate two and three-fifths multiplied by negative two-sevenths. Give your answer as a fraction in its simplest form.

Our first step in this question is to convert the mixed number two and three-fifths to a top heavy or improper fraction. In the bar shown, we have shaded two and three-fifths. Each complete bar is equal to five-fifths. This means that altogether, we have thirteen-fifths shaded. The fraction two and three-fifths is equal to thirteen-fifths. A quicker way of calculating this is to multiply the whole number by the denominator and then adding the numerator. Two multiplied by five is 10, and adding three gives us 13. This is the numerator of our improper fraction. We, therefore, need to multiply thirteen-fifths by negative two-sevenths.

We recall that when multiplying two fractions ๐ over ๐ and ๐ over ๐, we simply multiply the numerators and separately multiply the denominators. We also need to remember that when we multiply a positive number by a negative number, we get a negative answer. 13 multiplied by two is 26. Five multiplied by seven is 35. Multiplying positive thirteen-fifths by negative two-sevenths is equal to negative 26 over 35 or negative twenty-six thirty-fifths.

Our next question is a worded problem in context.

Sara works in a supermarket. She earns seven dollars per hour. How much will she get paid if she puts in 35 and one-quarter hours per week? Write your answer as a decimal.

Saraโs pay will be equal to her wage per hour multiplied by the number of hours she works. We are told she earns seven dollars per hour. We are also told that she works for 35 and a quarter hours per week. We need to multiply this by seven. There are lots of ways of working out this calculation. One way would be to multiply seven by 30, seven by five, and seven by a quarter. As seven multiplied by three is 21, seven multiplied by 30 is 210. Seven multiplied by five is 35. Seven multiplied by one-quarter is seven-quarters.

As seven divided by four is equal to one remainder three, seven-quarters is the same as the mixed number one and three-quarters. We know that three-quarters is equal to the decimal 0.75. Therefore, one and three-quarters is equal to 1.75. We need to add 210, 35, and 1.75. This is equal to 246.75. If Sara earns seven dollars per hour and works for 35 and a quarter hours, she will earn 246 dollars and 75 cents.

In our next question, we will multiply a percentage by a mixed number.

Calculate 50 percent of one and a half. Give your answer as a fraction.

In order to answer this question, we firstly need to convert 50 percent into a fraction. As the word โpercentโ means out of 100, we can convert from a percentage to a decimal by dividing by 100. 50 divided by 100 is 0.5. This is equal to five-tenths. We can then simplify this fraction by dividing the numerator and denominator by five. 50 percent is equal to one-half. Next, we need to convert 1.5 into a top heavy or improper fraction. There are two-halves in one whole one. Therefore, one and a half is equal to three-halves or three over two. We know that the word โofโ in mathematics means multiply. We need to multiply one-half by three-halves.

When multiplying two fractions, we multiply the two numerators and then the two denominators separately. One multiplied by three is three, and two multiplied by two is four. We can, therefore, conclude that 50 percent of one and a half is three-quarters. We could also have shown this pictorially. We began with one and a half and wanted to calculate 50 percent or a half of this. One-half of a whole one is equal to one-half, and one-half or 50 percent of a half is a quarter. Adding a half and a quarter once again gives us a final answer of three-quarters.

The penultimate question in this video involves multiplying a percentage by a decimal.

Calculate 25 percent multiplied by 0.2. Give your answer as a decimal number.

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. The word โpercentโ means out of 100. Therefore, 25 percent is equal to 25 over or out of 100. As the line in the fraction means divide, this is equal to 0.25. The fraction can also be simplified by dividing the numerator and denominator by 25. 25 percent is, therefore, equal to one-quarter. This means that we have two options to proceed. We can either multiply one-quarter by 0.2 or 0.25 by 0.2.

Multiplying by a quarter is the same as dividing by four. We need to divide 0.2 by four. This is equal to 0.05. When multiplying the two decimals 0.25 and 0.2, we know that 25 multiplied by two is 50. 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. 0.25 multiplied by 0.2 is equal to 0.050. This is the same as 0.05. 25 percent multiplied by 0.2 written as a decimal is 0.05.

Our final question involves multiplying an integer, a fraction, and a decimal.

Calculate 25 multiplied by one-sixth multiplied by 0.08. Give your answer as a fraction in its simplest form.

As we need to give our answer as a fraction, our first step is to convert 0.08 into a fraction. As the eight is in the hundredths column, this is equal to eight hundredths or eight over 100. Both the numerator and denominator are divisible by four, so this fraction simplifies to two over 25 or two twenty-fifths. We need to multiply 25, one-sixth, and two twenty-fifths. Any integer or whole number, in this case, 25, can be written over one. Before multiplying the fractions, we can now cross simplify or cross cancel.

There is a 25 on the numerator and denominator, so these will cancel. The numbers two and six are both even, so they are divisible by two. On the numerators, we are left with one multiplied by one multiplied by one. And on the denominator, we are left with one multiplied by three multiplied by one. 25 multiplied by one-sixth multiplied by 0.08 is, therefore, equal to one-third.

We will now summarize the key points from this video. 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. Rational numbers include positive and negative integers, fractions, and any recurring or terminating decimal. They also include percentages. Where there is a mixture of these, we need to convert them all into either fractions or decimals. When multiplying two or more fractions, we multiply the numerators and denominators separately. However, it is useful to cross cancel or cross simplify first. 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.

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