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Answer :
Let's tackle each question one at a time.
Q18: Finding the ratio P:Q:R:S:T
Given ratios:
- [tex]P:Q = 4:3[/tex]
- [tex]Q:R = 5:6[/tex]
- [tex]R:S = 9:8[/tex]
- [tex]T:S = 10:11[/tex]
We want to find a common ratio [tex]P:Q:R:S:T[/tex].
Step-by-step to find the ratios:
For [tex]P:Q = 4:3[/tex], we can set [tex]P = 4k[/tex] and [tex]Q = 3k[/tex].
For [tex]Q:R = 5:6[/tex], since [tex]Q = 3k[/tex], then [tex]R = \frac{6}{5} \times 3k = 3.6k[/tex].
For [tex]R:S = 9:8[/tex], [tex]S = \frac{8}{9} \times 3.6k = 3.2k[/tex].
For [tex]T:S = 10:11[/tex], [tex]T = \frac{10}{11} \times 3.2k = 2.909k[/tex].
Converting ratios to integers:
To find the smallest integer ratio, multiply through by the least common multiple of the denominators:
- Multiply all by 55 to clear out the fractions and to bring these to whole numbers:
[tex]P = 4k \times 55 = 220,
Q = 3k \times 55 = 165,
R = 3.6k \times 55 = 198,
S = 3.2k \times 55 = 176,
T = 2.909k \times 55 = 160[/tex]
Therefore, the ratio [tex]P:Q:R:S:T[/tex] is 220:165:198:176:160.
The correct option is C.
Q19: Arranging letters M, P, T, S, X, Z, F, E with E and F together
Treat E and F as a single unit (or block), so now you are arranging 7 units: M, P, T, S, X, Z, and (EF).
The 7 units can be arranged in [tex]7![/tex] ways.
[tex]7! = 5040[/tex] ways
Within the block EF, E and F can be arranged in [tex]2![/tex] ways.
Thus, total ways = [tex]7! \times 2! = 5040 \times 2 = 10080[/tex].
The correct option is A.
Q20: Finding the cyclist's actual speed without wind
The cyclist covers 10 km in 30 minutes against the wind.
The effective speed against the wind = [tex]x - 5[/tex] km/h
10 km in 0.5 hours implies [tex]x - 5 = \frac{10}{0.5} = 20[/tex]
Thus, [tex]x = 25[/tex] km/h
The correct option is D.
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