determine whether the number sentence is true. In each case explain how you could answer without calculating. Check your answers by doing the indicated calculations.
A. 50 x 432= (50 x 400) + (50 x 32)
B. 50 x 368 = (50 x 400) - (50 x 32)
C. 50 x 800= ( 50 x 1000) + (50 x 200)
D. (70 x 20) + ( 50 x 20)= 90 x 70
E. 50 + ( 400 x 32) = ( 50 + 400) x (50 + 32)
F. 6 x 17 = 6 x 20 - 6 x 3
Online, “*” is used to indicate multiplication to avoid confusion with “x” as an unknown.
A, 400 + 32 = 432
I'll let you do the calculations
A. To determine whether the number sentence is true, you can use the distributive property. The distributive property states that the product of a number and the sum of two other numbers is equal to the sum of the products of the number and each of the other two numbers.
In this case, the number sentence is 50 x 432 = (50 x 400) + (50 x 32). By using the distributive property, we can calculate the right-hand side of the equation: (50 x 400) + (50 x 32) = 20000 + 1600 = 21600.
Now, to check our answer, we can perform the actual calculation of 50 x 432: 50 x 432 = 21600. Since both the calculated value and the answer we obtained by applying the distributive property are the same, we can conclude that the number sentence is true.
B. Using the same logic as in the previous case, the number sentence 50 x 368 = (50 x 400) - (50 x 32) can be evaluated using the distributive property: (50 x 400) - (50 x 32) = 20000 - 1600 = 18400.
Performing the actual multiplication, we get: 50 x 368 = 18400. Again, both the calculated value and the value obtained by applying the distributive property are the same, indicating that the number sentence is true.
C. By using the distributive property, we can evaluate the equation 50 x 800 = (50 x 1000) + (50 x 200): (50 x 1000) + (50 x 200) = 50000 + 10000 = 60000.
Calculating the actual product, we have: 50 x 800 = 40000. Since the calculated value and the value obtained by applying the distributive property differ, we can conclude that the number sentence is false.
D. The equation (70 x 20) + (50 x 20) = 90 x 70 can be evaluated by performing the calculations on both sides: (70 x 20) + (50 x 20) = 1400 + 1000 = 2400.
Calculating the right side of the equation: 90 x 70 = 6300. Since the calculated value and the value obtained on the right-hand side of the equation are different, we can conclude that the number sentence is false.
E. Applying the distributive property to the equation 50 + (400 x 32) = (50 + 400) x (50 + 32):
(50 + 400) x (50 + 32) = 450 x 82 = 36900.
Performing the calculation, we get: 50 + (400 x 32) = 50 + 12800 = 12850.
Since the calculated value and the value obtained by applying the distributive property are different, we can conclude that the number sentence is false.
F. By applying the distributive property to the equation 6 x 17 = 6 x 20 - 6 x 3:
6 x 20 - 6 x 3 = 120 - 18 = 102.
Performing the actual multiplication, we get: 6 x 17 = 102.
Both the calculated value and the value obtained by applying the distributive property are the same, indicating that the number sentence is true.
By following these steps, you can determine whether a number sentence is true or false without performing the actual calculations. However, it's always important to double-check your answer by doing the indicated calculations to ensure accuracy.