Certainly! Let's simplify each ratio step-by-step.
### Part (a): Simplify [tex]\(0.4 \, \text{kg} : 320 \, \text{g}\)[/tex]
1. Convert kilograms to grams:
- [tex]\(0.4 \, \text{kg} \)[/tex] is equivalent to [tex]\(0.4 \times 1000 \, \text{g} = 400 \, \text{g}\)[/tex].
2. Form the ratio:
- The ratio now becomes [tex]\(400 \, \text{g} : 320 \, \text{g}\)[/tex].
3. Simplify the ratio:
- To simplify the ratio [tex]\(400 : 320\)[/tex], we divide both numbers by their greatest common divisor (GCD).
4. Calculate the GCD:
- The GCD of 400 and 320 is 80.
5. Divide both terms by the GCD:
- [tex]\(\frac{400}{80} : \frac{320}{80} = 5 : 4\)[/tex].
Thus, [tex]\(0.4 \, \text{kg} : 320 \, \text{g}\)[/tex] simplifies to:
[tex]\[ \frac{5}{4} \text{ or } 1.25 \][/tex]
### Part (b): Simplify [tex]\(0.12: 3.57\)[/tex]
1. Form the ratio:
- The given ratio is [tex]\(0.12 : 3.57\)[/tex].
2. Simplify the ratio:
- The first step in simplifying a ratio involving decimals is to express the ratio as a fraction.
- [tex]\(0.12 : 3.57\)[/tex] can be written as the fraction [tex]\(\frac{0.12}{3.57}\)[/tex].
3. Simplify the fraction:
- To simplify the fraction, find the equivalent fraction in its simplest form.
4. Convert the fraction to its simplest form:
- By finding the simplest form, we get [tex]\(\frac{4}{119}\)[/tex].
Thus, [tex]\(0.12 : 3.57\)[/tex] simplifies to:
[tex]\[ \frac{4}{119} \text{ or approximately } 0.03361344537815126 \][/tex]
And there you have the simplified forms of both ratios:
a) [tex]\(0.4 \, \text{kg} : 320 \, \text{g}\)[/tex] simplifies to [tex]\( \frac{5}{4} \text{ or } 1.25\)[/tex].
b) [tex]\(0.12 : 3.57\)[/tex] simplifies to [tex]\( \frac{4}{119} \text{ or approximately } 0.03361344537815126\)[/tex].
