The given ratios expresses the number of time a value is larger or smaller
than another value.
The correct responses are;
- 5.
![\underline{(E) \ \displaystyle \frac{11}{2}}](https://tex.z-dn.net/?f=%5Cunderline%7B%28E%29%20%5C%20%5Cdisplaystyle%20%20%5Cfrac%7B11%7D%7B2%7D%7D)
- 6.
![\underline{(E) \ \displaystyle \frac{5}{14}}](https://tex.z-dn.net/?f=%5Cunderline%7B%28E%29%20%5C%20%5Cdisplaystyle%20%5Cfrac%7B5%7D%7B14%7D%7D)
- 7. (B)
=<u> 6</u>,
= <u>9·√5</u>
- 8. <u>(C) The values are equal</u>
- 9. <u>(A) The value in column A is greater</u>
Reasons:
3. Given that the perimeter of the rectangle = 72
The ratio of the lengths of the sides = 1:2
Let <em>a</em> and <em>b</em> represent the sides, we have;
2·a + 2·b = 72
![\displaystyle \frac{a}{b} = \mathbf{\frac{1}{2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Ba%7D%7Bb%7D%20%3D%20%5Cmathbf%7B%5Cfrac%7B1%7D%7B2%7D%7D)
Which gives;
2·a = b
2·a + 2·(2·a) = 72
6·a = 72
a = 72 ÷ 6 = 12
b = 2·a = 2 × 12 = 24
The lengths of the sides are; <u>(B) 12 and 24</u>
4. Extended ratio = 2:3:7
The perimeter = 36
The lengths of the sides are;
![\displaystyle \frac{2}{2 + 3 + 7} \times 36 = \mathbf{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B2%7D%7B2%20%2B%203%20%2B%207%7D%20%5Ctimes%2036%20%3D%20%5Cmathbf%7B6%7D)
![\displaystyle \frac{3}{2 + 3 + 7} \times 36 = \mathbf{9}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B3%7D%7B2%20%2B%203%20%2B%207%7D%20%5Ctimes%2036%20%3D%20%5Cmathbf%7B9%7D)
![\displaystyle \frac{7}{2 + 3 + 7} \times 36 = \mathbf{21}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B7%7D%7B2%20%2B%203%20%2B%207%7D%20%5Ctimes%2036%20%3D%20%5Cmathbf%7B21%7D)
The lengths are; <u>(D) 6, 9, 21</u>
5. The given equation is presented as follows;
![\displaystyle \frac{5}{x + 7} = \mathbf{\frac{3}{x + 2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B5%7D%7Bx%20%2B%207%7D%20%3D%20%5Cmathbf%7B%5Cfrac%7B3%7D%7Bx%20%2B%202%7D%7D)
5 × (x + 2) = 3 × (x + 7)
5·x + 10 = 3·x + 21
5·x - 3·x = 21 - 10
2·x = 11
![\displaystyle x = \mathbf{ \frac{11}{2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%20%3D%20%5Cmathbf%7B%20%5Cfrac%7B11%7D%7B2%7D%7D)
The correct option is; ![\displaystyle \underline{(E) \ \frac{11}{2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cunderline%7B%28E%29%20%5C%20%5Cfrac%7B11%7D%7B2%7D%7D)
6. The width to length ratio is ![\displaystyle \mathbf{\frac{2.5}{7.0}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7B2.5%7D%7B7.0%7D%7D)
The simplified ratio is therefore;
![\displaystyle \frac{2.5}{7.0} = \frac{2 \times 2.5}{2 \times 7.0} = \mathbf{\frac{5}{14}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B2.5%7D%7B7.0%7D%20%3D%20%20%5Cfrac%7B2%20%5Ctimes%202.5%7D%7B2%20%5Ctimes%207.0%7D%20%3D%20%20%5Cmathbf%7B%5Cfrac%7B5%7D%7B14%7D%7D)
The correct option is (E) ![\displaystyle \underline{(E) \ \frac{5}{14}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cunderline%7B%28E%29%20%5C%20%5Cfrac%7B5%7D%7B14%7D%7D)
7. The given ratio of the lengths is 3:1
Therefore;
:
= 3:1
Which gives;
![\displaystyle \mathbf{\frac{\overline{BC}}{\overline{EF}}} = \frac{3}{1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7B%5Coverline%7BBC%7D%7D%7B%5Coverline%7BEF%7D%7D%7D%20%3D%20%5Cfrac%7B3%7D%7B1%7D)
= 18
Therefore;
![\displaystyle \frac{18}{\overline{EF}} = \frac{3}{1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B18%7D%7B%5Coverline%7BEF%7D%7D%20%3D%20%5Cfrac%7B3%7D%7B1%7D)
18 × 1 = 3 × ![\overline{EF}](https://tex.z-dn.net/?f=%5Coverline%7BEF%7D)
![\displaystyle \overline{EF} = \frac{18 \times 1}{3} = 6](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Coverline%7BEF%7D%20%3D%20%5Cfrac%7B18%20%5Ctimes%201%7D%7B3%7D%20%3D%206)
= 6
By Pythagorean theorem, we have;
² =
² +
²
Which gives;
² = 3² + 6² = 45
= √(45) = 3·√5
Using the given ratio, we have;
= 3 × ![\mathbf{\overline{DF}}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Coverline%7BDF%7D%7D)
Which gives;
= 3 × 3·√5 = 9·√5
= 9·√5
The correct option is; (B)
=<u> 6</u>,
= <u>9·√5</u>
8. EF = 1, AB = 2
CD = 2, CE = 4
Therefore;
![\displaystyle \mathbf{\frac{EF}{AB}} =\frac{1}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7BEF%7D%7BAB%7D%7D%20%3D%5Cfrac%7B1%7D%7B2%7D)
![\displaystyle \mathbf{\frac{CD}{CE}} =\frac{2}{4} = \frac{1}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7BCD%7D%7BCE%7D%7D%20%3D%5Cfrac%7B2%7D%7B4%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
Which gives;
![\displaystyle \frac{EF}{AB} =\displaystyle \mathbf{\frac{CD}{CE}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BEF%7D%7BAB%7D%20%3D%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7BCD%7D%7BCE%7D%7D)
<u>(C) The values are equal</u>
9. AC = 6, BE = 8
DF = 3, BD = 6
Column A
![\displaystyle \frac{AC}{BE} =\displaystyle \mathbf{\frac{6}{8}} = \frac{3}{4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BAC%7D%7BBE%7D%20%3D%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7B6%7D%7B8%7D%7D%20%3D%20%5Cfrac%7B3%7D%7B4%7D)
Column B
![\displaystyle \frac{DF}{BD} =\displaystyle \mathbf{\frac{3}{6}} = \frac{1}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7BDF%7D%7BBD%7D%20%3D%5Cdisplaystyle%20%5Cmathbf%7B%5Cfrac%7B3%7D%7B6%7D%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D)
![\displaystyle \frac{3}{4} > \mathbf{\frac{1}{2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B3%7D%7B4%7D%20%3E%20%5Cmathbf%7B%5Cfrac%7B1%7D%7B2%7D%7D)
Therefore;
Column A is greater than column B
<u>(A) The value in column A is greater</u>
Learn more about ratios here:
brainly.com/question/2774839