Discover new perspectives and gain insights with IDNLearn.com. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.
Sagot :
Sure, let's tackle these ratio and proportion problems step-by-step:
### Question 8
#### Part a)
Given:
1. [tex]\( p: q = 1: 2 \)[/tex]
2. [tex]\( q: r = 4: 3 \)[/tex]
We need to find [tex]\( r \)[/tex].
First, express both ratios with the same [tex]\( q \)[/tex].
1. From [tex]\( p: q = 1: 2 \)[/tex], we have:
[tex]\[ p = 1 \text{ unit}, \quad q = 2 \text{ units} \][/tex]
2. From [tex]\( q: r = 4: 3 \)[/tex], we have:
[tex]\[ q = 4 \text{ units}, \quad r = 3 \text{ units} \][/tex]
To combine these ratios, find a common value for [tex]\( q \)[/tex]. The least common multiple (LCM) of 2 and 4 is 4.
Scale the first ratio [tex]\( p: q \)[/tex]:
[tex]\[ p = 1 \text{ unit} \quad \text{scaled by factor of 2:} \quad p = 2 \text{ units} \][/tex]
[tex]\[ q = 2 \text{ units} \quad \text{scaled by factor of 2:} \quad q = 4 \text{ units} \][/tex]
So, [tex]\( p: q: r = 2: 4: 3 \)[/tex].
Thus, [tex]\( r \)[/tex] in terms of [tex]\( p \)[/tex] is [tex]\( r = 3 \)[/tex].
### Part b)
Given:
1. [tex]\( a: b = 2: 3 \)[/tex]
2. [tex]\( b: c = 6: 5 \)[/tex]
We need to find [tex]\( c \)[/tex].
First, express both ratios with the same [tex]\( b \)[/tex].
1. From [tex]\( a: b = 2: 3 \)[/tex], we have:
[tex]\[ a = 2 \text{ units}, \quad b = 3 \text{ units} \][/tex]
2. From [tex]\( b: c = 6: 5 \)[/tex], we have:
[tex]\[ b = 6 \text{ units}, \quad c = 5 \text{ units} \][/tex]
To combine these ratios, find a common value for [tex]\( b \)[/tex]. The least common multiple (LCM) of 3 and 6 is 6.
Scale the first ratio [tex]\( a: b \)[/tex]:
[tex]\[ a = 2 \text{ units} \quad \text{scaled by factor of 2:} \quad a = 4 \text{ units} \][/tex]
[tex]\[ b = 3 \text{ units} \quad \text{scaled by factor of 2:} \quad b = 6 \text{ units} \][/tex]
So, [tex]\( a: b: c = 4: 6: 5 \)[/tex].
Thus, [tex]\( c \)[/tex] in terms of [tex]\( a \)[/tex] is [tex]\( c = 5 \)[/tex].
### Question 9
#### Part a)
Given the proportion:
[tex]\[ x: 3 = 4: 6 \][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[ x \cdot 6 = 4 \cdot 3 \][/tex]
[tex]\[ 6x = 12 \][/tex]
[tex]\[ x = 2 \][/tex]
#### Part b)
Given the proportion:
[tex]\[ 2: x = 6: 9 \][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[ 2 \cdot 9 = 6 \cdot x \][/tex]
[tex]\[ 18 = 6x \][/tex]
[tex]\[ x = 3 \][/tex]
#### Part c)
Given the proportion:
[tex]\[ 5: 4 = x: 12 \][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[ 5 \cdot 12 = 4 \cdot x \][/tex]
[tex]\[ 60 = 4x \][/tex]
[tex]\[ x = 15 \][/tex]
### Question 10
#### Part a)
The problem statement is incomplete, so it's not possible to provide a solution.
#### Part b)
Given:
- The ratio of length to breadth is [tex]\( 4: 3 \)[/tex].
- The breadth is [tex]\( 75 \)[/tex] meters.
We need to find the length.
From the ratio:
[tex]\[ \frac{\text{length}}{\text{breadth}} = \frac{4}{3} \][/tex]
Given the breadth:
[tex]\[ \text{length} = \frac{4}{3} \times 75 \][/tex]
[tex]\[ \text{length} = 100 \][/tex]
Thus, the length of the ground is [tex]\( 100 \)[/tex] meters.
### Question 8
#### Part a)
Given:
1. [tex]\( p: q = 1: 2 \)[/tex]
2. [tex]\( q: r = 4: 3 \)[/tex]
We need to find [tex]\( r \)[/tex].
First, express both ratios with the same [tex]\( q \)[/tex].
1. From [tex]\( p: q = 1: 2 \)[/tex], we have:
[tex]\[ p = 1 \text{ unit}, \quad q = 2 \text{ units} \][/tex]
2. From [tex]\( q: r = 4: 3 \)[/tex], we have:
[tex]\[ q = 4 \text{ units}, \quad r = 3 \text{ units} \][/tex]
To combine these ratios, find a common value for [tex]\( q \)[/tex]. The least common multiple (LCM) of 2 and 4 is 4.
Scale the first ratio [tex]\( p: q \)[/tex]:
[tex]\[ p = 1 \text{ unit} \quad \text{scaled by factor of 2:} \quad p = 2 \text{ units} \][/tex]
[tex]\[ q = 2 \text{ units} \quad \text{scaled by factor of 2:} \quad q = 4 \text{ units} \][/tex]
So, [tex]\( p: q: r = 2: 4: 3 \)[/tex].
Thus, [tex]\( r \)[/tex] in terms of [tex]\( p \)[/tex] is [tex]\( r = 3 \)[/tex].
### Part b)
Given:
1. [tex]\( a: b = 2: 3 \)[/tex]
2. [tex]\( b: c = 6: 5 \)[/tex]
We need to find [tex]\( c \)[/tex].
First, express both ratios with the same [tex]\( b \)[/tex].
1. From [tex]\( a: b = 2: 3 \)[/tex], we have:
[tex]\[ a = 2 \text{ units}, \quad b = 3 \text{ units} \][/tex]
2. From [tex]\( b: c = 6: 5 \)[/tex], we have:
[tex]\[ b = 6 \text{ units}, \quad c = 5 \text{ units} \][/tex]
To combine these ratios, find a common value for [tex]\( b \)[/tex]. The least common multiple (LCM) of 3 and 6 is 6.
Scale the first ratio [tex]\( a: b \)[/tex]:
[tex]\[ a = 2 \text{ units} \quad \text{scaled by factor of 2:} \quad a = 4 \text{ units} \][/tex]
[tex]\[ b = 3 \text{ units} \quad \text{scaled by factor of 2:} \quad b = 6 \text{ units} \][/tex]
So, [tex]\( a: b: c = 4: 6: 5 \)[/tex].
Thus, [tex]\( c \)[/tex] in terms of [tex]\( a \)[/tex] is [tex]\( c = 5 \)[/tex].
### Question 9
#### Part a)
Given the proportion:
[tex]\[ x: 3 = 4: 6 \][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[ x \cdot 6 = 4 \cdot 3 \][/tex]
[tex]\[ 6x = 12 \][/tex]
[tex]\[ x = 2 \][/tex]
#### Part b)
Given the proportion:
[tex]\[ 2: x = 6: 9 \][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[ 2 \cdot 9 = 6 \cdot x \][/tex]
[tex]\[ 18 = 6x \][/tex]
[tex]\[ x = 3 \][/tex]
#### Part c)
Given the proportion:
[tex]\[ 5: 4 = x: 12 \][/tex]
Cross-multiply to solve for [tex]\( x \)[/tex]:
[tex]\[ 5 \cdot 12 = 4 \cdot x \][/tex]
[tex]\[ 60 = 4x \][/tex]
[tex]\[ x = 15 \][/tex]
### Question 10
#### Part a)
The problem statement is incomplete, so it's not possible to provide a solution.
#### Part b)
Given:
- The ratio of length to breadth is [tex]\( 4: 3 \)[/tex].
- The breadth is [tex]\( 75 \)[/tex] meters.
We need to find the length.
From the ratio:
[tex]\[ \frac{\text{length}}{\text{breadth}} = \frac{4}{3} \][/tex]
Given the breadth:
[tex]\[ \text{length} = \frac{4}{3} \times 75 \][/tex]
[tex]\[ \text{length} = 100 \][/tex]
Thus, the length of the ground is [tex]\( 100 \)[/tex] meters.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.
