Algebra and Arithmetic

Given  $a:b = 3:4$ and $b:c = 8:9$
Then $\displaystyle \frac{a}{b}\times \frac{b}{c}=\frac{3}{4}\times \frac{8}{9}$

The $a:b:c=A:B:C$
can be represented by ratio representation is
$ratio=\dfrac{a}{A}=\dfrac{b}{B}=\dfrac{c}{C}$

Given: $\dfrac{A}{B}=\dfrac{2}{3}$

To find A:D, multiply the ratios: (A/B) * (B/C) * (C/D) = (2/3) * (4/5) * (5/9). The B and C terms cancel out, leaving A/D = (2 * 4 * 5) / (3 * 5 * 9) = 40/135 = 8/27.

Given that,  If $A:B=3:4,B:C=5:6$,then$A:C$

Given$ a:b=3:4$  and $b:c= 5:6$  

$a:b::b:c =3:4::5:6$ 

$a:c= 3/4*5/6=5/8$ 

Hence, this is the answer. 

$b:c=\begin{pmatrix}6\times\dfrac{7}{6}\end{pmatrix}:\begin{pmatrix}11\times\dfrac{7}{6}\end{pmatrix}=7:\dfrac{77}{6}$
$a:b:c=5:7:\dfrac{77}{6}=30:42:77$

Multiplying -1 by itself an even number of times results in 1. Since 10 is an even number, the product is 1.

The product of two negative integers is always a positive integer (e.g., -2 * -3 = 6).

Calculating both sides: (-8) + (-4) = -12, while (-8) - (-4) = -8 + 4 = -4. Since -12 is less than -4, the statement (-8) + (-4) < (-8) - (-4) is true.

Subtracting a negative from a negative is equivalent to adding a positive. For example, -5 - (-2) = -3, but -2 - (-5) = +3. The result depends on the magnitudes.

By definition, all integers less than 0 are negative integers, which are denoted with a minus sign.

To find what must be subtracted, set up the equation: (2a + b) - x = (2a - b). Solving for x gives x = (2a + b) - (2a - b) = 2a + b - 2a + b = 2b.

When dividing a negative integer by another negative integer, the signs cancel out, resulting in a positive quotient.

(10A + 1) + (10 + B) = 10B + 0. Simplifying: 10A + B + 11 = 10B. 10A + 11 = 9B. Testing A=7: 70 + 11 = 81, so 9B = 81, B = 9. This matches.

Option C is an identity (2/3x + 2 = 2/3x + 2). Option A is incorrect because (x+2)^2 = x^2 + 4x + 4. Option B is incorrect because a^2 + b^2 is not (a+b)^2.