15 on cross multiplication

Cross Multiplication

In this section we look at the useful procedure of cross multiplcation.


In this section we look at the useful procedure of cross multiplcation.Cross Multiplication


In this section we look at the useful procedure of cross multiplcation.Cross Multiplication Many procedures with two fractions utilize the operation of cross– multiplication as shown below.



ab

cd

Cross Multiplication Many procedures with two fractions utilize the operation of cross– multiplication as shown below.



ab

cd

Cross Multiplication Many procedures with two fractions utilize the operation of cross– multiplication as shown below.

ad bc



ab

cd

Cross Multiplication Many procedures with two fractions utilize the operation of cross– multiplication as shown below. Take the denominators and multiply them diagonally across.

ad bc



What we get are two numbers.

ab

cd


ad bc




ab

cd


ad bcMake sure that the denominators cross over and up so the numerators stay put.




ab

cd


ad bcMake sure that the denominators cross over and up so the numerators stay put. Do not cross downward as shown here. a

bcdadbc


Here are some operations where we may cross multiply. Cross Multiplication

Here are some operations where we may cross multiply. Rephrasing Fractional Ratios


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2,


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2, and it’s written as 3 : 2 for sugar : flour.


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2, and it’s written as 3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is 2 : 3.


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2, and it’s written as 3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is 2 : 3. For most people a recipe that calls for the fractional ratio of 3/4 cup sugar to 2/3 cup of flour is confusing.


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2, and it’s written as 3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is 2 : 3. For most people a recipe that calls for the fractional ratio of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to cross multiply to rewrite this ratio in whole numbers.


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2, and it’s written as 3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is 2 : 3. For most people a recipe that calls for the fractional ratio of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to cross multiply to rewrite this ratio in whole numbers. Example A.rewrite a recipe that calls for the fractional ratio of 3/4 cup sugar to 2/3 cup of flour into ratio of whole numbers.


Here are some operations where we may cross multiply. Rephrasing Fractional RatiosIf a cookie recipe calls for 3 cups of sugar and 2 cups of flour, we say the ratio of sugar to flour is 3 to 2, and it’s written as 3 : 2 for sugar : flour. Or we said the ratio of flour : sugar is 2 : 3. For most people a recipe that calls for the fractional ratio of 3/4 cup sugar to 2/3 cup of flour is confusing. It’s better to cross multiply to rewrite this ratio in whole numbers. Example A.rewrite a recipe that calls for the fractional ratio of 3/4 cup sugar to 2/3 cup of flour into ratio of whole numbers.Write 3/4 cup of sugar as and 2/3 cup of flour as3

4 S 23 F.



4 S 23 F.

We have the ratio 34 S : 2

3 F



4 S 23 F.


3 F cross multiply we’ve 9S : 8F.



4 S 23 F.



Hence in integers, the ratio is 9 : 8 for sugar : flour.



4 S 23 F.



Hence in integers, the ratio is 9 : 8 for sugar : flour.


Remark: A ratio such as 8 : 4 should be simplified to 2 : 1.

Cross–Multiplication Test for Comparing Two Fractions Cross Multiplication


When comparing two fractions to see which is larger and which is smaller.


When comparing two fractions to see which is larger and which is smaller. Cross–multiply them, the side with the larger product corresponds to the larger fraction.


When comparing two fractions to see which is larger and which is smaller. Cross–multiply them, the side with the larger product corresponds to the larger fraction.In particular, if the cross multiplication products are the same then the fraction are the same.

Cross–Multiplication Test for Comparing Two Fractions

Hence cross– multiply



35

915





35

915

=45 45

we get





35

915

=45 45 so35

915=

we get





35

915

=45 45 so35

915=

we get

35

58





35

915

=45 45 so35

915=

we get

Cross– multiply 35

58





35

915

=45 45 so35

915=

we get


58

24 25we get





35

915

=45 45 so35

915=

we get


58

24 25we getmoreless





35

915

=45 45 so35

915=

we get


58

24 25

Hence 35

58is less than

we getmoreless

.





35

915

=45 45 so35

915=

we get


58

24 25

Hence 35

58is less than

we getmoreless

.

(Which is more 711

914 or ? Do it by inspection.)

Cross–Multiplication for Addition or SubtractionCross Multiplication

Cross–Multiplication for Addition or SubtractionWe may cross multiply to add or subtract two fractions



ab

cd±



ab

cd± = ad ±bc


Cross–Multiplication for Addition or Subtraction

ab

cd± = ad ±bc


We may cross multiply to add or subtract two fractions with the product of the denominators as the common denominator.

Cross–Multiplication for Addition or Subtraction

ab

cd± = ad ±bc

bd


We may cross multiply to add or subtract two fractions with the product of the denominators as the common denominator.

Cross–Multiplication for Addition or SubtractionWe may cross multiply to add or subtract two fractions with the product of the denominators as the common denominator.

ab

cd

Afterwards we reduce if necessary for the simplified answer.

± = ad ±bcbd



ab

cd

Afterwards we reduce if necessary for the simplified answer.Example B. Calculate

± = ad ±bcbd

35

56 – a.



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*5a.



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – b.



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*12b.



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b.



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b. 5

36=



ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b. 5

36=


In a. the LCD = 30 = 6*5 so the crossing method is the same as the Multiplier Method.


ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b. 5

36=


In a. the LCD = 30 = 6*5 so the crossing method is the same as the Multiplier Method. However in b. the crossing method yielded an answer that needed to be reduced.


ab

cd


± = ad ±bcbd

35

56 – = 5*5 – 6*3

6*57

30=a.

512

59 – =5*12 – 9*5

9*1215108=b. 5

36=


In a. the LCD = 30 = 6*5 so the crossing method is the same as the Multiplier Method. However in b. the crossing method yielded an answer that needed to be reduced. we need both methods.

Clearing Denominators–for Comparing Multiple Fractions Cross Multiplication

Cross multiplication of two fractions clears the denominators out of the picture when comparing them. To compare three or more fractions, we use their LCD to accomplish the same goal. (Multiplier Method for Comparing Fractions) To compare three or more fractions, multiply each fraction by the LCD first then compare their products.

23

Example C. Arrange the fractions from the smallest to the largest.

,35 ,

47

The LCD is 3(5)(7) = 105, multiply it to the fractions. Comparing the products, we see that from the smallest to the largest, they are

23,

35 ,

47

x 105

60,63, 70

23 ,

35,

47

the largestthe 2nd largestthe smallest

Ex. Restate the following ratios in integers.

9. In a market, ¾ of an apple may be traded with ½ a pear.Restate this using integers.

12

13 :1. 2. 3. 4.2

312 : 3

413 : 2

334 :

35

12 :5. 6. 7. 8.1

617 : 3

547 : 5

274 :

Determine which fraction is more and which is less.23

34 ,10. 11. 12. 13.4

534 , 4

735 , 5

645 ,

59

47 ,14. 15.

16. 17.7

1023 , 5

1237 , 13

885 ,

12

13 +18. 19. 20. 21.1

213 – 2

332 + 3

425 +

56

47 – 22. 23.

24. 25.7

1025 – 5

1134 + 5

97

15 –


C. Use cross–multiplication to combine the fractions.

15 on cross multiplication

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