Given that a : b : c = 75 : 120 : 132,. simplify a : b : c,. find b : a,. find b : c with full explanation and simplification ( if used)
The Correct Answer and Explanation is:
We are given the ratio: a:b:c=75:120:132a : b : c = 75 : 120 : 132
Step 1: Simplify the ratio a : b : c
To simplify a ratio, we divide all parts of the ratio by their greatest common divisor (GCD).
Find the GCD of 75, 120, and 132:
- Prime factors of 75: 3×523 \times 5^2
- Prime factors of 120: 23×3×52^3 \times 3 \times 5
- Prime factors of 132: 22×3×112^2 \times 3 \times 11
The common factor in all three numbers is 3.
So, divide each part by 3: 753:1203:1323=25:40:44\frac{75}{3} : \frac{120}{3} : \frac{132}{3} = 25 : 40 : 44
Final Simplified Ratio:
a:b:c=25:40:44a : b : c = 25 : 40 : 44
Step 2: Find b : a
From the simplified ratio: a:b=25:40⇒b:a=40:25a : b = 25 : 40 \Rightarrow b : a = 40 : 25
Simplify: 405:255=8:5\frac{40}{5} : \frac{25}{5} = 8 : 5
Answer:
b:a=8:5b : a = 8 : 5
Step 3: Find b : c
From the simplified ratio: b:c=40:44b : c = 40 : 44
Simplify: 404:444=10:11\frac{40}{4} : \frac{44}{4} = 10 : 11
Answer:
b:c=10:11b : c = 10 : 11
Explanation:
Ratios are a way of comparing quantities relative to each other. In the problem, we are given the ratio a:b:c=75:120:132a : b : c = 75 : 120 : 132. This is a compound ratio involving three parts. To work with such a ratio, the first step is to simplify it, just as we would simplify a fraction.
To simplify the three-part ratio, we find the greatest common divisor (GCD) of the three numbers. The GCD of 75, 120, and 132 is 3, since all three numbers share 3 as their highest common factor. Dividing each term by 3, we get the simplified ratio: 25:40:4425 : 40 : 44. This tells us that for every 25 units of a, there are 40 units of b and 44 units of c.
Next, to find the ratio of b to a, we reverse the order of a : b and write it as b : a, which becomes 40:2540 : 25. To simplify, we divide both numbers by their GCD, which is 5, giving us 8:58 : 5.
To find the ratio of b to c, we use the values from the simplified form: b:c=40:44b : c = 40 : 44. Their GCD is 4, so simplifying gives 10:1110 : 11.
Understanding and simplifying ratios helps us compare different quantities efficiently. It is widely used in real-life contexts such as mixing ingredients, budgeting, and scaling measurements in science and engineering.
