# a. (69+45)+55b. 4×324×32c. 4×(16×25)4×(16×25)d. (250+86)+50

a. $4u-3v-w=0$
b. $4×324×32$
c. $4×\left(16×25\right)4×\left(16×25\right)$
d. $\left(250+86\right)+50$

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To multiply and add numbers mentally using the addition and multiplication properties, we want to rearrange the numbers so we get sums or products that are multiples of 10.
a. Since 45 and 55 add to a multiple of 10, we first use the Associative Property $\left(a+b\right)+c=a+\left(b+c\right)\left(a+b\right)+c$ to get $\left(69+45\right)+55=69+\left(45+55\right)$ since $45+55=100$
Using mental math we then get $69+\left(45+55\right)=69+100=169$
b. Since $32=30+2$, we can write $4×32=4×\left(30+2\right)$. Using the Distributive Property $a\left(b+c\right)=ab+ac$ then gives $4×\left(30+2\right)=4×30+4×2$. Using mental math we then get $4×30+4×2=120+8=128.$
c. Since $4×25=100$, we want to rearrange the product so 4 and 25 are next to each other. Using the Commutative Property $ab=ba$ to switch the 16 and 25 gives $4×\left(16×25\right)=4×\left(25×16\right).$
Using the Associative Property $a\left(bc\right)=\left(ab\right)c$ then gives $4×\left(25×16\right)=\left(4×25\right)×16$.
Using mental math we then get $\left(4×25\right)×16=100×16=1600$.
d. Since $250+50=300$, we want to rearrange the sum so 250 and 50 are next to each other. Using the Commutative Property $a+b=b+a$ to switch the 250 and 86 gives
$\left(250+86\right)+50=\left(86+250\right)+50$.
Using the Associative Property $\left(a+b\right)+c=a+\left(b+c\right)\left(a+b\right)+c=a+\left(b+c\right)$ then gives $\left(86+250\right)+50=86+\left(250+50\right)$.
Using mental math we then get $86+\left(250+50\right)=86+300=386$.