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Homework Progress
31/43 Marks
The prices of two radios are in the ratio x : y
When the prices are both increased by £20, the ratio becomes 5:2
When the prices are both reduced by £5, the ratio becomes 5:1
Express the ratio x: y in its lowest terms.
Optional working
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Sagot :

sqdancefan

Answer:

  x : y = 4 : 1

Step-by-step explanation:

You want x:y, given adding 20 to both makes the ratio 5:2, and subtracting 5 from both makes the ratio 5:1.

Setup

The given relations tell us ...

  (x +20) : (y +20) = 5 : 2

  (x -5) : (y -5) = 5 : 1

Solution

Cross multiplying gives the equations ...

  2(x +20) = 5(y +20) . . . . . . . [eq1]

  (x -5) = 5(y -5) . . . . . . . . . . . . [eq2]

Solving [eq2] for x gives ...

  x = 5y -20

Substituting that into [eq1], we have ...

  2(5y -20 +20) = 5(y +20)

  10y = 5y +100

  5y = 100

  y = 20

  x = 5y -10 = 100 -20 = 80

Then x:y = 80:20 = 4:1.

The ratio x:y in lowest terms is 4:1.

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Additional comment

When working with ratios, sometimes it works well to consider the difference of the numerator and denominator. Here, the difference when 20 is added is 5-2 = 3 "ratio units", which is 3/2 times the value y+20. The difference when 5 is subtracted is 5-1 = 4 "ratio units", which is 4 times the value y-5. Adding or subtracting the same number does not change the actual difference (x-y). This tells us ...

  (x -y) = 3/2(y+20) = 4(y-5)
  3y +60 = 8y -40
  100 = 5y
  20 = y
  x = y + 4(y-5) = 20 +4(15) = 80

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