Luna is thinking of a fraction equivalent to \frac{7}{8}. The numerator is greater than 23 and the denominator is less than 35. What fraction is Luna thinking of?
The Correct Answer and Explanation is:
To find the fraction Luna is thinking of, we are given:
- It is equivalent to 78\frac{7}{8}87
- The numerator is greater than 23
- The denominator is less than 35
Let us begin by understanding what it means for a fraction to be equivalent to 78\frac{7}{8}87. Two fractions are equivalent if they represent the same value. Mathematically, this happens when one fraction is a multiple of the other. So we can multiply both the numerator and denominator of 78\frac{7}{8}87 by the same whole number.
Let us multiply both 7 and 8 by various whole numbers and see when the numerator becomes greater than 23 and the denominator stays under 35.
Start with:
- 7×1=77 \times 1 = 77×1=7, 8×1=88 \times 1 = 88×1=8
- 7×2=147 \times 2 = 147×2=14, 8×2=168 \times 2 = 168×2=16
- 7×3=217 \times 3 = 217×3=21, 8×3=248 \times 3 = 248×3=24
- 7×4=287 \times 4 = 287×4=28, 8×4=328 \times 4 = 328×4=32
- 7×5=357 \times 5 = 357×5=35, 8×5=408 \times 5 = 408×5=40 ← Denominator too big
So the only multiple where the numerator is greater than 23 and the denominator is less than 35 is:2832\frac{28}{32}3228
Now let us verify that 2832\frac{28}{32}3228 is indeed equivalent to 78\frac{7}{8}87. Simplify 2832\frac{28}{32}3228:
- The greatest common divisor of 28 and 32 is 4
- Divide both by 4: 28÷4=728 \div 4 = 728÷4=7, 32÷4=832 \div 4 = 832÷4=8
So:2832=78\frac{28}{32} = \frac{7}{8}3228=87
Thus, Luna is thinking of the fraction 2832\frac{28}{32}3228.
Explanation
Luna is trying to think of a fraction that is equivalent to 78\frac{7}{8}87 but with specific conditions: the numerator must be greater than 23, and the denominator must be less than 35. To solve this, we start by exploring equivalent fractions. Two fractions are equivalent if they represent the same value. This happens when both the numerator and denominator are multiplied by the same number. For example, if we multiply both 7 and 8 by 2, we get 1416\frac{14}{16}1614, which is equivalent to 78\frac{7}{8}87.
We continue this pattern and generate the following equivalent fractions: 1416\frac{14}{16}1614, 2124\frac{21}{24}2421, 2832\frac{28}{32}3228, and so on. Each step involves multiplying both 7 and 8 by an increasing whole number. We stop when the numerator exceeds 23 and the denominator is still below 35.
Upon examining the options, 2832\frac{28}{32}3228 is the only one that fits the criteria: the numerator is 28, which is greater than 23, and the denominator is 32, which is less than 35. To confirm that it is truly equivalent to 78\frac{7}{8}87, we simplify 2832\frac{28}{32}3228 by dividing both numbers by their greatest common divisor, which is 4. This gives us 78\frac{7}{8}87, confirming the match.
Therefore, the fraction Luna is thinking of is 2832\frac{28}{32}3228, which meets both conditions and is mathematically equivalent to 78\frac{7}{8}87.
