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# help...

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Three runners Adam, Ben, and Charles all start at the same time for a 24 km race, and each of them runs at a constant speed. When Adam finishes the race, Ben is 8 km behind, and Charles is 12 km behind. When Ben finishes the race, how far behind is Charles, in km?

Jan 18, 2020

#1
+24133
+3

Three runners Adam, Ben, and Charles all start
at the same time for a 24 km race,
and each of them runs at a constant speed.
When Adam finishes the race, Ben is 8 km behind, and Charles is 12 km behind.
When Ben finishes the race, how far behind is Charles, in km?

Formula $$s=v*t$$

$$\begin{array}{|lrcll|} \hline (1) & 24~\text{km} &=& v_{Adam}*t_{Adam} \\ (2) & 24~\text{km}-8~\text{km} &=& v_{Ben}*t_{Adam} \\ &\mathbf{ v_{Ben}} &=& \mathbf{\dfrac{16}{t_{Adam}} } \\\\ (3) & 24~\text{km}-12~\text{km} &=& v_{Charles}*t_{Adam} \\ & \mathbf{v_{Charles}} &=& \mathbf{ \dfrac{12}{t_{Adam}} } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline s_{Ben}=24~\text{km} &=& v_{Ben}*t_{Ben} \quad |\quad \mathbf{ v_{Ben}=\dfrac{16}{t_{Adam}} } \\ \\ 24 &=& \dfrac{16}{t_{Adam}}*t_{Ben} \\\\ \dfrac{t_{Ben}}{t_{Adam}} &=& \dfrac{24}{16} \\ \\ \mathbf{ \dfrac{t_{Ben}}{t_{Adam}} } &=& \mathbf{1.5 } \\\\ \hline s_{Charles} &=& v_{Charles}*t_{Ben} \quad |\quad \mathbf{v_{Charles}=\dfrac{12}{t_{Adam}} } \\ \\ s_{Charles} &=& \dfrac{12}{t_{Adam}}*t_{Ben} \\ \\ s_{Charles} &=& 12*\dfrac{t_{Ben} }{t_{Adam}} \quad | \quad \mathbf{ \dfrac{t_{Ben}}{t_{Adam}} = 1.5 } \\\\ s_{Charles} &=& 12* 1.5 \\\\ s_{Charles} &=& 18~\text{km} \\ \hline s_{Ben}-s_{Charles} &=& 24~\text{km}-18~\text{km}\\ \mathbf{s_{Ben}-s_{Charles}} &=& \mathbf{6~\text{km}} \\ \hline \end{array}$$

Charles is $$\mathbf{6~\text{km}}$$ behind

Jan 18, 2020
#2
+20428
+4

OK... lets say adam runs at 24 km /hr  finishes in 1 hour

charles is 12 km behind .....he is only 1/2 way through....must be running  at 12 km/hr   he will finish in 2 hours

ben is 8km behind    he has run 16 km in one hour     16 km/hr    he will finish in 1  1/2 hour

Charles  done in 2 hours

whne Ben finishes Charles is   still  30 minutes behind x 12/km/hr  = 6 km

Jan 18, 2020
#3
+8974
+2

Three runners Adam, Ben, and Charles all start at the same time for a 24 km race, and each of them runs at a constant speed. When Adam finishes the race, Ben is 8 km behind, and Charles is 12 km behind. When Ben finishes the race, how far behind is Charles, in km?

Drei Läufer, Adam, Ben und Charles, starten alle gleichzeitig für ein 24-km-Rennen, und jeder von ihnen läuft mit einer konstanten Geschwindigkeit. Als Adam das Rennen beendet hat, liegt Ben 8 km und Charles 12 km zurück. Wenn Ben das Rennen beendet, wie weit ist Charles zurück.

Hello Guest!

$$v=\frac{s}{t}\\ t=\frac{s}{v}\\ s=vt$$

$$\frac{24}{v_a}=\frac{16}{v_b}=\frac{12}{v_c}$$

$$\frac{v_b}{16}=\frac{v_c}{12}\\ v_c=\frac{3}{4}v_b$$

$$\frac{24km}{v_b}=\frac{24km-x}{v_c}\\ \frac{24km}{v_b}=\frac{24km-x}{\frac{3}{4}v_b}\\ 3\cdot 24km=4(24km-x)\\ 72km=96km-4x\\ 4x=(96-72)km$$

$$x=6km$$

Charles is 6km behind.

!

Jan 18, 2020
#4
+2

Thanks for all the help! :)

Jan 18, 2020