Examkin

Soru

Zorluk: OrtaFractions and Rational Numbers

If aa and bb are positive rational numbers such that a<b<1a < b < 1, which of the following expressions must be strictly greater than 11? Select all that apply.

  1. ba\frac{b}{a}Cevap
  2. a+1b\frac{a+1}{b}Cevap
  3. C
    ab2\frac{a}{b^2}
  4. b+1a+1\frac{b+1}{a+1}Cevap
  5. E
    a+b2b\frac{a+b}{2b}

Cevap

The expressions that must be strictly greater than 1 are the ratio of b to a, the fraction (a + 1) over b, and the fraction (b + 1) over (a + 1).
The correct options are those involving ratios where the numerator is demonstrably greater than the denominator. The ratio of b to a compares two positive numbers where b is larger, giving a quotient greater than 1. The fraction with numerator (a + 1) has a value over 1 while denominator b is under 1, ensuring the quotient exceeds 1. The fraction comparing (b + 1) to (a + 1) has a larger numerator because adding 1 preserves the inequality b > a.

Adım Adım Çözüm

1
Analyze the ratio of b to a
Since b>a>0b > a > 0, ba>1\frac{b}{a} > 1 is always true.
Dividing any positive number by a smaller positive number results in a value greater than 1.
2
Analyze the fraction (a + 1) over b
Since a>0a > 0, a+1>1a + 1 > 1. Since b<1b < 1, a+1>ba + 1 > b, which implies a+1b>1\frac{a+1}{b} > 1.
The numerator is greater than 1 and the denominator is less than 1, so the fraction exceeds 1.
3
Test counterexamples for the fraction a over b squared
For a=18a = \frac{1}{8} and b=12b = \frac{1}{2}, ab2=1/81/4=12≤1\frac{a}{b^2} = \frac{1/8}{1/4} = \frac{1}{2} \le 1.
The expression is not guaranteed to be greater than 1 for all valid values of a and b.
4
Analyze the fraction (b + 1) over (a + 1)
Since b>ab > a, adding 1 to both quantities yields b+1>a+1>0b + 1 > a + 1 > 0, so b+1a+1>1\frac{b+1}{a+1} > 1.
Adding the same positive quantity to two numbers maintains their relative order.
5
Analyze the average fraction (a + b) over 2b
Since a<ba < b, a+b<2ba + b < 2b, meaning a+b2b<1\frac{a+b}{2b} < 1.
The numerator is the sum of a and b, which is strictly less than twice b.

Anahtar Kavram

Properties of Rational Number Inequalities and Fraction Comparison
Tahmini Süre:1m 30s
Bu soruyu puanla

Konular

Fractions and Rational Numbers

Bu konudan daha fazla soru