Question

what is the next fraction in this sequence? simplify your answer.
$\frac{4}{5}$, $\frac{3}{4}$, $\frac{7}{10}$, $\frac{13}{20}$, ...

Explanation:

Step1: Find common denominator

Convert fractions to 20 denominator:
$\frac{4}{5}=\frac{16}{20}$, $\frac{3}{4}=\frac{15}{20}$, $\frac{7}{10}=\frac{14}{20}$, $\frac{13}{20}$ remains.

Step2: Analyze numerator pattern

Numerators: 16, 15, 14, 13. Decrease by 1? Wait, no—wait, check differences: 16 to 15: -1, 15 to 14: -1, 14 to 13: -1? Wait, no, wait the original sequence: $\frac{4}{5},\frac{3}{4},\frac{7}{10},\frac{13}{20}$—wait, maybe I messed up. Wait, let's recalculate:

Wait, $\frac{4}{5} = 0.8$, $\frac{3}{4}=0.75$, $\frac{7}{10}=0.7$, $\frac{13}{20}=0.65$. Ah! So each term decreases by 0.05? Wait, 0.8 - 0.05 = 0.75 ($\frac{3}{4}$), 0.75 - 0.05 = 0.7 ($\frac{7}{10}$), 0.7 - 0.05 = 0.65 ($\frac{13}{20}$). So next term: 0.65 - 0.05 = 0.6, which is $\frac{3}{5}$? Wait, no, wait 0.6 is $\frac{12}{20}$? Wait, no, wait my first approach was wrong. Wait, let's list numerators and denominators separately.

Wait, denominators: 5, 4, 10, 20. Let's find a pattern. 5, 4, 10, 20. Multiply by 2.5? No. Wait, convert to 20: 54=20, 45=20, 102=20, 201=20. So denominators are 20 when converted. Numerators: 44=16, 35=15, 72=14, 131=13. Wait, 16,15,14,13: each decreases by 1. So next numerator: 13 -1 =12. So fraction is $\frac{12}{20}=\frac{3}{5}$? Wait, but 0.65 -0.05=0.6, which is 3/5. Wait, but let's check the differences between terms.

Alternative approach: Find the difference between consecutive fractions.

$\frac{3}{4} - \frac{4}{5} = \frac{15}{20} - \frac{16}{20} = -\frac{1}{20}$

$\frac{7}{10} - \frac{3}{4} = \frac{14}{20} - \frac{15}{20} = -\frac{1}{20}$

$\frac{13}{20} - \frac{7}{10} = \frac{13}{20} - \frac{14}{20} = -\frac{1}{20}$

Ah! So each term is previous term minus $\frac{1}{20}$. So next term: $\frac{13}{20} - \frac{1}{20} = \frac{12}{20} = \frac{3}{5}$.

Step3: Simplify the fraction

$\frac{12}{20}$ simplifies by dividing numerator and denominator by 4: $\frac{12\div4}{20\div4}=\frac{3}{5}$.

Answer:

$\frac{3}{5}$