32 Question 2 Write the following fractions in their simplest form. \( 2.1 \frac{2}{10}=- \) \( 2.2 \quad \frac{40}{200}=- \) \( 2.3 \quad 3 \frac{2}{5}=- \) \( 2.42,25=- \) \( 5 \cdot 0,13 \dot{5}=- \)

We are asked to “write the following fractions in their simplest form.” (Notice that in many European countries the comma “,” is used as the decimal separator.) In the list below the numbers before the fraction (2.1, 2.2, etc.) are not parts of the numbers but are labels for the items. Below are the five items and the step‐by‐step simplification: ────────────────────────────── Item 2.1: 2/10 Start with: 2/10 • Divide numerator and denominator by their greatest common divisor, 2.   2 ÷ 2 = 1  and  10 ÷ 2 = 5. Thus, 2/10 simplifies to 1/5. ────────────────────────────── Item 2.2: 40/200 Start with: 40/200 • Note that 40 divides both numerator and denominator.   Divide by 40: 40 ÷ 40 = 1  and  200 ÷ 40 = 5. Thus, 40/200 simplifies to 1/5. ────────────────────────────── Item 2.3: 3 2/5 This is a mixed number. • One may leave the answer as a mixed number if desired, but usually we write it as an improper fraction. • To convert, multiply the whole number by the denominator and add the numerator:   3 × 5 = 15, then 15 + 2 = 17. Thus, 3 2/5 = 17/5. ────────────────────────────── Item 2.4: 2,25 Here 2,25 means 2.25 (using a comma as the decimal point). • Write 2.25 as a fraction:   2.25 = 225/100. • Now simplify by dividing numerator and denominator by their greatest common divisor, 25:   225 ÷ 25 = 9  and  100 ÷ 25 = 4. Thus, 2,25 simplifies to 9/4. ────────────────────────────── Item 2.5: 5 · 0,13˙5 Here 0,13˙5 means 0.13 with a repeating 5. In other words, the number is 0.135555… where only the digit 5 repeats. Step 1. Convert 0.13˙5 to a fraction. Let x = 0.135555… The repeating part (5) begins after the hundredths place. A standard method is as follows: • Since 0.13 is non‐repeating (2 digits) and there is 1 repeating digit, multiply x by 100 (to shift the non‐repeating digits) and by an extra factor 10 (for the one repeating digit):   Multiply by 100: 100x = 13.5555…   Multiply by 10: 10·(100x) = 1000x = 135.5555… • Now subtract the two equations to eliminate the repeating part:   1000x − 100x = (135.5555… − 13.5555…)   900x = 122 • Solve for x:   x = 122/900   Simplify by dividing numerator and denominator by 2:   122 ÷ 2 = 61  and  900 ÷ 2 = 450,   so x = 61/450. Step 2. Multiply by 5:   5 · x = 5 · (61/450) = 305/450. Now simplify 305/450 by dividing numerator and denominator by 5:   305 ÷ 5 = 61  and  450 ÷ 5 = 90. Thus, 5 · 0,13˙5 simplifies to 61/90. ────────────────────────────── Summary of Answers:  • Item 2.1: 2/10 = 1/5  • Item 2.2: 40/200 = 1/5  • Item 2.3: 3 2/5 = 17/5  • Item 2.4: 2,25 = 9/4  • Item 2.5: 5 · 0,13˙5 = 61/90 These are the fractions written in their simplest form.