Combining the fixed whole number 5 with each of the 6 valid proper fractions gives 6 distinct mixed numbers: 5(1)/(2), 5(1)/(3), 5(1)/(7), 5(2)/(3), 5(2)/(7), and 5(3)/(7). All 6 numbers lie strictly between 5 and 6.

Combining the fixed whole number 5 with each of the 6 valid proper fractions gives 6 distinct mixed numbers: 5(1)/(2), 5(1)/(3), 5(1)/(7), 5(2)/(3), 5(2)/(7), and 5(3)/(7). All 6 numbers lie strictly between 5 and 6.

How many mixed numbers between the value 5 and 6 can be made using the following cards: 5, 1 ,7, 2, 3

Answer: 6

Step-by-step solution

Step 1: Understand the structure of the mixed number

A mixed number consists of a whole number part $W$ and a proper fractional part $\frac{N}{D}$ , where $N < D$ . For the mixed number to lie strictly between 5 and 6, its whole number part must be 5. Since each card can be used at most once, the card 5 is fixed as the whole number part.

Step 2: Identify available cards for numerator and denominator

With the card 5 fixed as the whole number part $W$ , the remaining available cards for the numerator $N$ and denominator $D$ are 1, 2, 3, and 7. The fractional part $\frac{N}{D}$ must be a proper fraction, meaning $N < D$ , and the denominator cannot be 0.

Step 3: Count the valid proper fractions

To form a proper fraction $\frac{N}{D}$ , we need to choose 2 distinct digits from $\{1, 2, 3, 7\}$ such that $N < D$ . Choosing any pair of 2 digits uniquely determines the fraction because the smaller digit must be the numerator $N$ and the larger digit must be the denominator $D$ . The number of ways to choose 2 digits from 4 is given by $^4C_2 = \frac{4 \times 3}{2} = 6$ .

Step 4: List all possible mixed numbers

Combining the fixed whole number 5 with each of the 6 valid proper fractions gives 6 distinct mixed numbers: $5\frac{1}{2}$ , $5\frac{1}{3}$ , $5\frac{1}{7}$ , $5\frac{2}{3}$ , $5\frac{2}{7}$ , and $5\frac{3}{7}$ . All 6 numbers lie strictly between 5 and 6.

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