Byron wants to use the distributive property to rewrite the addition problem shown so that the numbers left in the parentheses have no common factor except 1.
(48 + 36)
Which is an equivalent expression that has numbers in the parentheses whose only common factor is 1?
A.
4(12 + 9)
B.
2(24 + 18)
C.
12(4 + 3)
D.
36(12 + 1)
B. 2(24 + 18)
Mr. Hon purchases a new car every 4 years. Ms. Jasper purchases a new car every 6 years. They both purchased new cars this year. When will they next both purchase new cars in the same year?
A.
in 2 years
B.
in 8 years
C.
in 12 years
D.
in 24 years
The next time Mr. Hon purchases a new car will be in 4 years from now. The next time Ms. Jasper purchases a new car will be in 6 years from now. To find the next time they both purchase new cars in the same year, we need to find the least common multiple (LCM) of 4 and 6.
The prime factorization of 4 is 2^2.
The prime factorization of 6 is 2 * 3.
To find the LCM, we take the highest power of each prime factor that appears in either number. In this case, the LCM is 2^2 * 3 = 12.
Therefore, the next time they both purchase new cars in the same year will be in 12 years.
Answer: C. in 12 years
Which expression is NOT equivalent to 9x+4y+13-3x-y-10 ?
A. 3(2x+y+1)
B. 6x+3(1+y)
C. (9-3)x+(4-0)y+3
D. (13-10)+(9x-3x)+(4y-y)
D. (13-10)+(9x-3x)+(4y-y)
To rewrite the addition problem (48 + 36) using the distributive property so that the numbers left in the parentheses have no common factor except 1, you need to find expressions where the greatest common factor between each term outside the parentheses and each term inside the parentheses is 1.
Let's evaluate each option to see which one satisfies this condition.
A. 4(12 + 9)
The greatest common factor between 4 and both 12 and 9 is 1, so this expression satisfies the condition.
B. 2(24 + 18)
The greatest common factor between 2 and both 24 and 18 is 2, so this expression does not satisfy the condition.
C. 12(4 + 3)
The greatest common factor between 12 and both 4 and 3 is 1, so this expression satisfies the condition.
D. 36(12 + 1)
The greatest common factor between 36 and both 12 and 1 is 1, so this expression satisfies the condition.
Therefore, both options A. 4(12 + 9) and C. 12(4 + 3) are equivalent expressions that have numbers in the parentheses whose only common factor is 1.
To rewrite the addition problem (48 + 36) using the distributive property, we need to find an equivalent expression where the numbers in the parentheses have no common factor except 1.
The distributive property states that a(b + c) = ab + ac, where a, b, and c are any numbers. This property allows us to distribute the multiplication to each term inside the parentheses.
Let's examine the options:
A. 4(12 + 9)
B. 2(24 + 18)
C. 12(4 + 3)
D. 36(12 + 1)
In option A, we can distribute the 4 to both terms inside the parentheses: 4(12) + 4(9) = 48 + 36. However, the numbers in the parentheses, 12 and 9, have a common factor of 3, so this option does not satisfy the requirement of having no common factors except 1.
In option B, we can distribute the 2 to both terms inside the parentheses: 2(24) + 2(18) = 48 + 36. Similarly to option A, the numbers in the parentheses, 24 and 18, have a common factor of 6, so this option is also not the correct choice.
In option C, we can distribute the 12 to both terms inside the parentheses: 12(4) + 12(3) = 48 + 36. Here, the numbers in the parentheses, 4 and 3, have no common factors except 1, so option C satisfies the requirement.
In option D, we can distribute the 36 to both terms inside the parentheses: 36(12) + 36(1) = 48 + 36. However, the numbers in the parentheses, 12 and 1, have a common factor of 1, so this option does not meet the requirement.
Therefore, the equivalent expression that has numbers in the parentheses whose only common factor is 1 is option C: 12(4 + 3).