Question Video: Using Properties of Continued Proportions to Simplify an Algebraic Fraction Mathematics • 7th Grade

If 𝑏 is the middle proportion between 𝑎 and 𝑐, then which of the following is equal to (𝑎² + 𝑏²)/(𝑏² + 𝑐²)? [A] 𝑐/𝑎 [B] 𝑎/𝑐 [C] 2𝑐/𝑎 [D] 2𝑎/𝑐

02:31

Video Transcript

If 𝑏 is the middle proportion between 𝑎 and 𝑐, then which of the following is equal to 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared? Is it option (A) 𝑐 over 𝑎, option (B) 𝑎 over 𝑐, option (C) two 𝑐 over 𝑎, or option (D) two 𝑎 over 𝑐?

We will begin by recalling the definition of the middle proportion. We know that if 𝑏 is the middle proportion between 𝑎 and 𝑐, then 𝑎 over 𝑏 is equal to 𝑏 over 𝑐. In this question, we are given an expression involving 𝑎 squared, 𝑏 squared, and 𝑐 squared, and we need to decide which of the four options given are equal to this. We notice that none of these options contain 𝑏. The equation 𝑎 over 𝑏 is equal to 𝑏 over 𝑐 can be rewritten as 𝑎𝑐 is equal to 𝑏 squared.

We can now substitute the expression for 𝑏 squared into our expression 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared. This becomes 𝑎 squared plus 𝑎𝑐 over 𝑎𝑐 plus 𝑐 squared. The two terms on the numerator have a common factor of 𝑎. We can therefore factor that this out, giving us 𝑎 multiplied by 𝑎 plus 𝑐 since 𝑎 multiplied by 𝑎 is 𝑎 squared and 𝑎 multiplied by 𝑐 is 𝑎𝑐. The two terms on the denominator have a common factor of 𝑐. Factoring this out gives us 𝑐 multiplied by 𝑎 plus 𝑐. Since both the numerator and denominator have a common factor of 𝑎 plus 𝑐, we can cancel this, leaving us with a simplified expression, 𝑎 over 𝑐.

If 𝑏 is the middle proportion between 𝑎 and 𝑐, then 𝑎 squared plus 𝑏 squared over 𝑏 squared plus 𝑐 squared is equal to 𝑎 over 𝑐. Of the four options listed, the correct answer is option (B).

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.