4 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




ab

cd


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




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.



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 as34 S 2

3 F.





3 F.

We have the ratio 34 S : 2

3 F





3 F.


3 F cross multiply we’ve 9S : 8F.





3 F.



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





3 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 25

we get





35

915

=45 45 so35

915=

we get


58

24 25

we get

moreless





35

915

=45 45 so35

915=

we get


58

24 25

Hence 35

58is less than

we get

moreless

.





35

915

=45 45 so35

915=

we get


58

24 25

Hence 35

58is less than

we get

moreless

.

(Which is more 711

914 or ? Do it by inspection.)

Multiplying by the LCDCross Multiplication


Cross–multiplication is a shortcut for clearing denominators for two fractions.


Cross–multiplication is a shortcut for clearing denominators for two fractions. When there are more than two fractions, we use their LCD to clear their denominators when needed.



Example B.A cookie recipe that calls for 1/4 cup of butter, 2/3 cup of flourand 5/6 cup of sugar. Phrase this in terms of whole number.




56 S

We have the three–way ratio:14 B: 2

3 F:




56 S

14 B: 2

3 F:( )12

We have the three–way ratio: multiply the list by the LCD 12.




56 S

We have the three–way ratio: multiply the list by the LCD 12.14 B: 2

3 F:( )123 4 2

3B: 8F: 10S




56 S


3 F:( )123 4 2

3B: 8F: 10S

Hence recipe calls for 3: 8: 10 for butter: flour: sugar.




56 S


3 F:( )123 4 2

3B: 8F: 10S


Example C. Arrange 3/5, 2/3 and 7/12 from the smallest to the largest.




56 S


3 F:( )123 4 2

3B: 8F: 10S



712

35

: 23 :Multiply the list by the LCD = 60.




56 S


3 F:( )123 4 2

3B: 8F: 10S



712

35

: 23 :( ) 60Multiply the list by the LCD = 60.




56 S


3 F:( )123 4 2

3B: 8F: 10S



712

35

: 23 :( ) 60

36: 40: 35

12 520

Multiply the list by the LCD = 60.




56 S


3 F:( )123 4 2

3B: 8F: 10S



Multiply the list by the LCD = 60. 712

35

: 23 :( ) 60

36: 40: 357

1235

23 Hence <<

12 520

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


Example D. Calculate

± = ad ±bcbd

35

56 – a.



ab

cd



± = 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 – 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 b. the crossing method gives an answer that needs to be reduced because the denominator 9*12 =108 is not the LCD.

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.

4 cross multiplication

Education

denominators cross

ad bc cross multiplication

b c d cross multiplication