Problem Solving

In a race of 600 metres, Adam beats Victor by 60 metres. In a race of 500 metres, Victor beats Angie by 25 metres. By how many metres will Adam beat Angie in a 400 metre race?

OA 58 Metres

Detailed explanations would be appreciated. Many thanks in advance.

knight247 wrote:In a race of 600 metres, Adam beats Victor by 60 metres. In a race of 500 metres, Victor beats Angie by 25 metres. By how many metres will Adam beat Angie in a 400 metre race?

OA 58 Metres
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Detailed explanations would be appreciated. Many thanks in advance.

In a 600-meter race, Adam travels 600 meters in the time that Victor takes to travel 540 meters, with the result that Adam beats Victor by 60 meters.
Thus, if Adam and Victor travel for the same amount of time, their distances are in the following ratio:
(Adam's distance) : (Victor's distance) = 600:540 = 10:9.

In a 500-meter race, Victor travels 500 meters in the time that Angie takes to travel 475 meters, with the result that Victor beats Angie by 25 meters.
Thus, if Victor and Angie travel for the same amount of time, their distances are in the following ratio:
(Victor's distance) : (Angie's distance) = 500:475 = 20:19.

To combine the ratios, the element common to both ratios -- Victor's distance -- must be represented the SAME VALUE in each ratio.
Since Victor is represented by 9 in the first ratio and by 20 in the second ratio, convert the ratios so that Victor is represented by 9*20 = 180 in each ratio:
(Adam's distance) : (Victor's distance) = 10:9 = (10*20) : (9*20) = 200:180.
(Victor's distance) : (Angie's distance) = 20:19 = (20*9) : (19*9) = 180:171.

Combining the ratios, we get:
(Adam's distance) : (Victor's distance) : (Angie's distance) = 200:180:171.
Implication:
For every 200 meters Adam travels, Angie will travel 171 meters.
Thus, if Adam travels 2*200 = 400 meters, Angie will travel 2*171 = 342 meters.
Result:
In a 400-meter race, Adam will beat Angie by 400-342 = 58 meters.

knight247 wrote:In a race of 600 metres, Adam beats Victor by 60 metres. In a race of 500 metres, Victor beats Angie by 25 metres. By how many metres will Adam beat Angie in a 400 metre race?

OA 58 Metres

Detailed explanations would be appreciated. Many thanks in advance.

We don't know the times for any of this, but we know the distances covered in the same time.

In the same amount of time, Adam goes 600 meters and Victor is 60 meters behind at 540 meters.

Let's call Adam's rate A and Victor's V.

A x Time = 600
V x Time = 540

You could calculate it or, because the time is the same, just realize that A/V must be 600/540 = 10/9.

Next we get the ratio of Angie's speed, call that G, to Victor's speed, V. Keep Victor's speed in the bottom in both formulas because we know eventually we have to compare Adam's speed to Angie's.

Using the same logic, even though again we don't know the time, we know this.

G x Time = 475, V x Time = 500 and G/V = 475/500 = 19/20.

To get to the answer we need the ratio of Adam's speed to Angie's speed. We have everything set up to get that.

A/G = (A/V)/(G/V)

A/G = (10/9)/(19/20) = (10/9)*(20/19) = 200/171

So when Adam has gone 200 metres he will be 200 - 171 = 29 ahead of Angie, and when he goes double that, 400 metres, he will be 2 * 29 = 58 metres ahead of Angie.

Choose 58.

I think I have an intuitive way that uses our familiar rate formulas. (It looks long, but I just wanted to make the algebra explicit.)

Let's say that a = Adam's rate, v = Victor's rate, and b = Angie's rate. We'll say that t = the time it takes Victor to run 540 meters (i.e. the distance he's able to run in the race against Adam).

Since Adam runs 60 meters further than Victor in this time, we have 60 = (a - v)t.

The second equation is tricky. In the 500 meter race, Victor runs for (500/540) of the time he ran in the first place, which reduces to (25/27)t. So our equation for the second race is 25 = (v - b)(25/27)t.

Now we need to solve our final equation. The time here is (2/3)t because t happens to also represent the time it takes Adam to run 600 meters, and in this race we only have Adam running (2/3) of that.

d = (a - b) * (2/3)t

We want to find d. Now let's solve our three equations.

60 = at - vt, or vt = at - 60
25 = (25/27)vt - (25/27)bt

The second one quickly simplifies to vt = 27 - bt.

Since at - 60 and 27 - bt each equal vt, they also equal each other. This gives us

at - 60 = 27 - bt
at - bt = 87
t*(a - b) = 87.

We want (a - b)* t * (2/3), so multiply both sides by (2/3), which gives 87 * (2/3), or 58.

Here's a shorter but trickier way as well:

In our first race, Adam's D : Victor's D = 600 : 540 = 10 : 9. So Adam's distance over any amount of time is (10/9) * Victor's.

In our second race, Victor's D : Angie's D = 500 : 475 = 20 : 19. So Victor's distance over any amount of time is (20/19) * Angie's.

Let's say that Adam's D = x and Angie's D = y. Our equations above give

x = (10/9)*(20/19)*y

or

x = (200/171) * y

So x/y = 200/171. Since we want x to travel 400, we multiply both parts of the ratio by 2, giving 400/342. (400 - 342) = 58, so Adam beat Angie by 58 meters.

One last way that is "easier" (shorter, at least) than the first two I posted:

To figure out the relative position of the runners, let's have them all run a 600 meter race.

Adam wins. At the moment he hits the 600 meter mark, Victor is at the 540 meter mark.

Now here's the trick. If we extend Victor and Angie's race to 540 meters, each runner will be have run (54/50) of their original distance. So we can find Angie's distance at this point: it's 475 * (54/50). (This simplifies nicely: (25*19) * (54/50) = (19*54)/2 = 19*27 = 513.)

So when Adam is at the 600 meter mark, Angie is at the 513 meter mark. Now let's scale this down to a 400 meter race. 600 * (2/3) = 400, so we'll multiply each runner's distance by (2/3). This gives us Adam's D = 400 and Angie's D = (2/3)*513 = 342, so Adam wins by 58 meters.