The additive identity property says that a number does not change when zero is added to that number.
\(2.5+0=2.5\)
Examples:
\((1)\;4.2+0=4.2\\ (2)\;8.9+0=8.9\\ (3)\;11.52+0=11.52\)
The multiplicative identity property says that a number does not change when \(1\) is multiplied to that number.
Examples:
\(2.5×1=2.5\)
\(4.2×1=4.2\)
\(6.8×1=6.8\)
A \(12.2+0=12.2\)
B \(12.2×0=0\)
C \(12.2+1=13.2\)
D \(12.2+12.2=24.4\)
For addition: \(1.2+2.5=2.5+1.2\)
For multiplication: \(1.2×2.5=2.5×1.2\)
\(1.6+1.8=3.4\)
or
\(1.8+1.6=3.4\)
Thus, the answer is still the same.
Examples:
\((1)\;2.14×1.28=1.28×2.14\\ (2)\;3.15+7.45=7.45+3.15\\ (3)\;10.2×5.5=5.5×10.2\)
A \(5.5×6.5=5.5×6.5\)
B \(5.5+6.5=6.5+5.5\)
C \(5.5+6.5=5.5+6.5\)
D \(5.5×6.5=6.5×2.5\)
Associative property of addition:
\((1.2+2.3)+3.4=1.2+(2.3+3.4)\)
Associative property of multiplication:
\((1.2×2.3)3.4=1.2(2.3×3.4)\)
\((3.2+5.6)+4.1=3.2+(5.6+4.1)\)
\(8.8+4.1=3.2+9.7\)
\(12.9=12.9\)
In both the groupings, the sum of the addition does not change.
A \(6.2+(4.2+3.4)=6.2+(4.2+3.4)\)
B \(6.2+(4.2+3.4)=(6.2+3.4)+3.4\)
C \(6.2+(4.2+3.4)=(6.2+4.2)+3.4\)
D \((6.2+4.2)+3.4=6.2+(4.2+4.2)\)
Distributive property over addition:
The distributive property over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together.
\(4.2(3.1+1.2)=(4.2×3.1)+(4.2×1.2)\)
\(4.2×4.3=13.02+5.04\)
\(18.06=18.06\)
Distributive property over subtraction:
Either we find out the difference first and then multiply, or we first multiply with each number and then subtract, the result remains the same.
\(4.2(3.1-1.2)=(4.2×3.1)-(4.2×1.2)\)
\(4.2×1.9=13.02-5.04\)
\(7.98=7.98\)
\(4.2(3.1-1.2)\neq(4.2×1.2)-(4.2×3.1)\)
A \(8.4(2.7+4.8)\)
B \(2.7(3.6)\)
C \(2.7(8.4+4.8)\)
D \(2.7(8.4)+2.7(4.8)\)
An implicit expression can be solved as an explicit expression with the help of properties.
\(1.5=1+0.5\)
\(1.5(2.5+3.5)=1.5(2.5)+1.5(3.5)\)
\(1.5+(2.5+3.5)=(1.5+2.5)+3.5\)
\(1.5+2.5=2.5+1.5\)
For example: Solving \(800×25.5\)
\((25+.5)\)
\(800(25+.5)\)
\(800(25)+800(.5)\)
\(20000+400\)
\(=20,400\)