To write the comparison of 40 nickels to 22 dimes as a ratio in lowest terms, we can follow these steps:
1. Understand the values of nickels and dimes:
- One nickel is worth 5 cents.
- One dime is worth 10 cents.
2. Convert the quantities to their cent values:
- The total value of 40 nickels:
[tex]\[
40 \times 5 = 200 \text{ cents}
\][/tex]
- The total value of 22 dimes:
[tex]\[
22 \times 10 = 220 \text{ cents}
\][/tex]
3. Write the initial ratio:
- The ratio of the value in cents of nickels to dimes is:
[tex]\[
\frac{200 \text{ cents}}{220 \text{ cents}}
\][/tex]
4. Simplify the ratio by finding the greatest common divisor (GCD):
- The GCD of 200 and 220 can be calculated. The prime factorizations can help:
[tex]\[
200 = 2^3 \times 5^2
\][/tex]
[tex]\[
220 = 2^2 \times 5 \times 11
\][/tex]
- The common prime factors are [tex]\(2^2 \times 5\)[/tex], so:
[tex]\[
\text{GCD}(200, 220) = 20
\][/tex]
5. Divide both the numerator and the denominator by their GCD:
- Simplified numerator:
[tex]\[
200 \div 20 = 10
\][/tex]
- Simplified denominator:
[tex]\[
220 \div 20 = 11
\][/tex]
6. Express the simplified ratio:
- The ratio reduced to lowest terms is:
[tex]\[
\frac{10}{11}
\][/tex]
Therefore, the comparison of 40 nickels to 22 dimes, when reduced to the lowest terms, is [tex]\(10:11\)[/tex].
