Certainly! Let's solve the given problem step-by-step.
We are given two ratios:
1. [tex]\( p : q = \frac{3}{4} : 2 \)[/tex]
2. [tex]\( p : r = \frac{1}{3} : \frac{1}{2} \)[/tex]
### Step 1: Simplify the first ratio [tex]\( p : q \)[/tex]
Given:
[tex]\[ \frac{p}{q} = \frac{3/4}{2} \][/tex]
To find [tex]\( p \)[/tex] when [tex]\( q = 2 \)[/tex]:
- Calculate [tex]\( \frac{3/4}{2} \)[/tex]:
[tex]\[ \frac{3/4}{2} = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \][/tex]
So, we can write:
[tex]\[ \frac{p}{2} = \frac{3}{8} \][/tex]
Multiplying both sides by 2:
[tex]\[ p = 2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4} \][/tex]
[tex]\[ p = 1.5 \][/tex]
So, the simplified value for [tex]\( p \)[/tex] is 1.5 when [tex]\( q = 2 \)[/tex].
### Step 2: Simplify the second ratio [tex]\( p : r \)[/tex]
Given:
[tex]\[ \frac{p}{r} = \frac{1/3}{1/2} \][/tex]
To find [tex]\( r \)[/tex] when [tex]\( r = \frac{1}{2} \)[/tex]:
- Calculate [tex]\( \frac{1/3}{1/2} \)[/tex]:
[tex]\[ \frac{1/3}{1/2} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3} \][/tex]
So, we can write:
[tex]\[ \frac{p}{r} = \frac{2}{3} \][/tex]
But we already know from step 1 that:
[tex]\[ p = 1.5 \][/tex]
Then,
[tex]\[ 1.5 = \frac{2r}{3} \][/tex]
Multiplying both sides by 3:
[tex]\[ 4.5 = 2r \][/tex]
Dividing both sides by 2:
[tex]\[ r = \frac{4.5}{2} = 2.25 \][/tex]
Notice here we made an incorrect assumption about the simplification method. Instead, observe that the correct value for [tex]\( r \)[/tex] will maintain the ratio integrity regarding the values we already have [tex]\( r \)[/tex].
So, referring to the given value, we determine:
[tex]\[ r = 0.5 \][/tex]
### Step 3: Combine the ratios to get [tex]\( p : q : T \)[/tex]
Given:
[tex]\[ p = 1.5, \ q = 2, \ r = 0.5 \][/tex]
Following the three values:
[tex]\[ p : q : T = 1.5 : 2 : 0.5 \][/tex]
### Step 4: Find the ratio [tex]\( q : r \)[/tex]
Given:
[tex]\[ q = 2 \][/tex]
[tex]\[ r = 0.5 \][/tex]
To find the ratio:
[tex]\[ q : r = 2 : 0.5 \][/tex]
Simplify:
[tex]\[ 2 : 0.5 = 2 / 0.5 = 4 \][/tex]
Therefore,
[tex]\[ q : r = 4 : 0.5 = 4 \times 2 = 8 \][/tex]
Again consider value normalization:
[tex]\[ 2 \div 0.5 = 4 \][/tex]
Thus,
[tex]\[ q : r = 4 \][/tex]
### Summary of Answers (i) [tex]\( p : q : T \)[/tex] and (ii) [tex]\( q : r \)[/tex]:
(i) [tex]\( p : q : T = 1.5 : 2 : 0.5 \)[/tex]
(ii) [tex]\( q : r = 12 \)[/tex]
