Question 1147611
Matt's age now = M
James's age now = J
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<i>In 5 years, the sum of Matt and James age will be 100.</i>
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(M + 5) + (J + 5) = 100
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M + J + 10 = 100
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M + J = 90
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J = 90 - M
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So, revised:
Matt's age now = M 
James's age now = J = 90 - M
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<i>Matt is half as old as James will be when Matt is twice as old as James is now.</i>
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Let the period of time "when Matt is twice as old as James now" be equal to X.
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Look at the part of the statement where it says: "when Matt is twice as old as James is now."
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M + X = 2(90 - M)
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M + X = 180 - 2M
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3M + X = 180
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Now, look at the part of the statement where it says: "Matt is half as old as James will be when..."
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M = 0.5(90 - M + X)
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M = 45 - 0.5M + 0.5X
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1.5M - 0.5X = 45
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Now, you have two equations:
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3M + X = 180
1.5M - 0.5X = 45
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Multiply the second equation by 2:
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2(1.5M - 0.5X) = 2(45)
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3M - X = 90
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Here are your two equations now:
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3M + X = 180
3M - X = 90
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Add these two equations together and solve for M:
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6M = 270
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M = 45
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From earlier:
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Matt's age now = M = <b>45</b>
James's age now = J = 90 - M = 90 - 45 = <b>45</b>
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So, both men are 45.
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Check it:
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<i>In 5 years, the sum of Matt and James age will be 100.</i>
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Both men are 45 now, so in 5 years, both men will be 50.  This adds up to 100.
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<i>Matt is half as old as James will be when Matt is twice as old as James is now.</i>
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To check this, you need to figure out what X is.  Take either of our two equations:
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3M + X = 180
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We know M = 45, so:
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3(45) + X = 180
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135 + X = 180
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X = 45
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Let's look at the statement again:
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<i>Matt is half as old as James will be when Matt is twice as old as James is now.</i>
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"When Matt is twice as old as James is now" happens in 45 years.  (Because X = 45.)  In 45 years, Matt will be 90 and James is now 45.  That's twice as old.
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"Matt is half as old as James will be when..." Matt is half as old as James will be WHEN.  WHEN is the key word.  That WHEN is in 45 years.  So, again...Matt is half as old (45) as James will be in 45 years. In 45 years, James will be 90.  So, that's half as old.