Question

(a) in a classroom, there are 12 boys and 16 girls. what is the simplified ratio of boys to girls ( b:g )? (5 points)
(b) given ( a:b = 1:2 ) and ( b:c = 4:8 ), find ( a:b:c ) and then ( a:c ). (5 points)
(c) show that ( 2:6:7 ) is in proportion with ( 7:21:24.5 ). (5 points)
(d) if the ratio ( 1:y:2 ) and the ratio ( 14:21:28 ) are in proportion, then find the value of ( y ). (5 points)

Explanation:

Part (a)

Step1: Identify the numbers for ratio

We have 12 boys and 16 girls, so the ratio of boys to girls is \(12:16\).

Step2: Simplify the ratio

Find the greatest common divisor (GCD) of 12 and 16. The factors of 12 are \(1, 2, 3, 4, 6, 12\) and factors of 16 are \(1, 2, 4, 8, 16\). The GCD is 4. Divide both numbers by 4: \(\frac{12}{4}:\frac{16}{4}=3:4\).

Step1: Make the value of \(b\) consistent

Given \(a:b = 1:2\) and \(b:c=4:8\). We need to make the value of \(b\) same in both ratios. In \(a:b = 1:2\), multiply both parts by 2 to get \(a:b=2:4\) (since we want \(b = 4\) to match the second ratio).

Step2: Find \(a:b:c\)

Now, \(a:b = 2:4\) and \(b:c = 4:8\), so \(a:b:c=2:4:8\). We can simplify this by dividing by 2, so \(a:b:c = 1:2:4\).

Step3: Find \(a:c\)

From \(a:b:c=1:2:4\), the ratio of \(a\) to \(c\) is \(1:4\).

Step1: Check the ratio between corresponding terms

To check if two ratios \(2:6:7\) and \(7:21:24.5\) are in proportion, we check if \(\frac{7}{2}=\frac{21}{6}=\frac{24.5}{7}\).

Step2: Calculate each fraction

• \(\frac{7}{2}=3.5\)
• \(\frac{21}{6}=3.5\)
• \(\frac{24.5}{7}=3.5\)

Since all the fractions are equal, the two ratios are in proportion.

Answer:

\(3:4\)

Part (b)