Which of the following has the greatest value?

We can create a common denominator of (5^5)(7^5) for each answer choice. The choice with the largest numerator will be the fraction with the greatest value:

A)

[1/(5^3)(7^3)][(5^2)(7^2)/(5^2)(7^2)]

Numerator = (5^2)(7^2) = 1225

B)

[6/(5^4)(7^3)][(5^1)(7^2)/(5^1)(7^2)]

Numerator = (6)(5^1)(7^2) = 1470

C)

[45/(5^4)(7^4)][(5^1)(7^1)/(5^1)(7^1)]

Numerator = (45)(5^1)(7^1) = 1575

D)

[180/(5^5)(7^4)][(7^1)/(7^1)]

Numerator = (180)(7^1) = 1260

E)

Numerator = 900

Answer: C

Use the number properties we know:

Whenever we multiply a positive number by another number greater than 1, the number increases. Whenever we multiply it by a number less than 1, the number decreases.
e.g.

2 * (1/2) = 1 (multiplying by 1/2, decreases 2 to 1)

(1/3) * 2 = (2/3) (multiplying by 2 increases 1/3 to 2/3)

A. \(\frac{1}{5^37^3}\)

B. \(\frac{6}{5^47^3}\)

\(= \frac{6}{5} * (A)\) so this is greater than option (A). Eliminate (A)

C. \(\frac{45}{5^47^4}\)

\(= \frac{7.5}{7} * (B)\) so this is greater than option (B). Eliminate (B)

D. \(\frac{180}{5^57^4}\)

\(= \frac{4}{5} * (C)\) so this is less than option (C). Eliminate (D)

E. \(\frac{900}{5^57^5}\)

\(= \frac{20}{35} * (C)\) so this is less than option (C). Eliminate (E)

Answer (C)

Take the value presented in A as the base

A) 1
B) increase by 6/5
C) increase by 45/35 = 9/7
D) increase by 180/175
E) decrease by 900/1225

C has the largest increase in change

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