MHB Word problem: distance and profit

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The discussion focuses on solving two word problems involving distance and profit. For problem 30, the equations derived show the relationships between the distances traveled by individuals A, B, and C, leading to a formula for the time taken by B and C. The analysis reveals that the time taken can be expressed in terms of variables representing speed and distance. In problem 18, the total number of apples bought at different rates is calculated, leading to the conclusion that 44 apples were purchased. The participants seek guidance on these mathematical problems, emphasizing the need for clarity in the solution process.
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What I have tried for prob 30

Let $k+t=$total time taken by A
$t=$ total time taken by B and C

since $D_{A}=D_{B}=D_{C}$

$(k+t)c=(c+d)t$ solving for $t$

$t=\frac{ck}{d}$ hours

$\frac{c^2dk+cdk}{d}$ miles.

For 18, I don't know how to really start.

Please guide through correct solution for both problems. Thanks a lot!
 

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30.) I would let $x$ be the number of hours after $B$ started walking that $C$ began walking. Since $A$ and $B$ walked the same distance $D$, we may state:

$$D=ct=(c+d)(t-k)$$

$$ct=ct-ck+dt-dk$$

$$t=\frac{k(c+d)}{d}$$

Since $A$ and $C$ also walked the same distance, we have:

$$D=ct=(c+2d)(t-(k+x))=ct-c(k+x)+2dt-2d(k+x)$$

$$t=\frac{(c+2d)(k+x)}{2d}$$

Equating the two expressions for $t$, we obtain:

$$\frac{k(c+d)}{d}=\frac{(c+2d)(k+x)}{2d}$$

$$2k(c+d)=(c+2d)(k+x)$$

$$k+x=\frac{2k(c+d)}{c+2d}$$

$$x=\frac{2k(c+d)}{c+2d}-k=k\left(\frac{2(c+d)-(c+2d)}{c+2d)}\right)=\frac{ck}{c+2d}$$

$$D=\frac{ck(c+d)}{d}$$
 
18.) Let's let $N$ be the total number of apples bought, where:

$$N_3=$$ the number bought at 3 per cent.

$$N_4=\frac{5}{6}N_3=$$ the number bought at 4 per cent.

Hence:

$$N=N_3+N_4=\frac{11}{6}N_3$$

So, the total amount $A$ paid (in cents) is:

$$A=\frac{1}{3}N_3+\frac{1}{4}N_4=\frac{1}{3}N_3+\frac{5}{24}N_3=\frac{13}{24}N_3=\frac{13}{44}N$$

The amount received from selling $S$ is then:

$$S=\frac{3}{8}N=\frac{13}{44}N+\frac{7}{2}$$

And so we find:

$$N=44$$
 
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