Question

10. Weiming and Joyce intend to buy a present for Siti. Weiming agrees to pay at least twice as much as but at most $150 more than Joyce. Given that the present costs no more than $210, what is the greatest amount Weiming has to pay?​

Answer 1

Answer:

  $180

Step-by-step explanation:

The problem asks us to find the maximum value that Weiming must pay. Weiming is paying more than Joyce, and the total of their payments will be the cost of the present. Hence both will be paying the maximum amount when the present's cost is at its maximum: $210. If w represents the amount Weiming pays in that case, then the amount Joyce pays is 210-w.

The problem statement tells us two things about what Weiming pays. It sets a minimum of 2 times what Joyce pays, and it sets a maximum of 150 more than what Joyce pays. The problem requests the maximum Weiming pays, so we're only interested in the limit on the maximum.

  w ≤ 150 +(210 -w) . . . . . Weiming pays at most 150 more than Joyce pays

  2w ≤ 360 . . . . . add w, collect terms

  w ≤ 180 . . . . . divide by 2

Weiming pays at most $180.

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Additional comment

If we consider the more general case where the cost of the present is "c", then Joyce pays (c-w) and the relations become ...

  2(c -w) ≤ w ≤ 150 +(c -w) . . . . Weiming pays between twice Joyce's cost and 150 more than Joyce's cost

  2c -2w ≤ w ≤ 150 +c -w . . . . . eliminate parentheses

This is better solved by writing it as two inequalities:

  2c -2w ≤ w   ⇒   2c ≤ 3w   ⇒   2/3c ≤ w

  w ≤ 150 +c -w   ⇒   2w ≤ 150 +c   ⇒   w ≤ 75 +c/2

That is, the amount Weiming pays will be ...

  2/3c ≤ w ≤ 75 +c/2 . . . . . . between 2/3 the present cost and $75 more than half the present cost.

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Some compound inequalities can be solved by manipulating the entire expression. Here, we cannot isolate w to the middle expression when we keep the compound inequality together, so we must divide it into its parts to solve it.

Answer 2

The  required amount Weiming has to pay is given by the limit of $150

more than the amount Joyce has to pay.

Response:

• The greatest amount Weiming has to pay is $180

Which method can be used to analyze the amount paid by Weiming and Joyce?

Amount Weiming agrees to pay ≥ 2 × Joyce agrees to pay

Amount Weiming agrees to pay ≤ 150 + Amount Joyce agrees to pay

Let x represent the amount Weiming agrees to pay, and let y represent

the amount Joyce agrees to pay, we have;

x + y = 210

x ≥ 2·y

x ≤ y + 150

When the amount Weiming is $150 more than Joyce, which is the

limiting amount for Weiming, we have;

x = y + 150

Which gives;

y + 150 + y = 210

2·y = 210 - 150 = 60

y = 60 ÷ 2 = 30

x = 30 + 150 = 180

Therefore;

• The greatest amount Weiming has to pay is, x = $180

Learn more about inequalities here:

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