Activity Dylan, Alan, Bruce, and Cecil are brothers. Dylan weighs d pounds. Alan weighs 3 pounds more than twice what Dylan weighs. Bruce weighs 12 pounds less than three times what Dylan weighs. If Cecil weighs the same as the combined weight of Bruce and Alan, what is Cecil’s weight in terms of Dylan’s weight (d)?

Complete the steps below to solve this problem and others like it.

Part A
Alan weighs 3 pounds more than twice what Dylan weighs. Write an expression to represent Alan’s weight in terms of Dylan’s weight.

Part B
Bruce weighs 12 pounds less than three times what Dylan weighs. Write an expression for Bruce's weight in terms of Dylan's weight.\

Part C
Cecil weighs the same as the combined weight of Bruce and Alan. Use the expressions you found in parts A and B to write an expression for Cecil's weight in terms of Dylan’s weight.

Part D
Assume that Dylan weighs 45 pounds. Use the expression from part C to calculate Cecil’s weight.

Part E
Recall the linear expression for Cecil’s weight in part D. Before plugging values into it, could you have simplified the expression? If yes, explain how.

Part F
Simplify the expression (2d + 3) + (3d − 12).

Part G
Dylan weighs 45 pounds. Use the simplified expression you found in Part F to recalculate Cecil’s weight. Show your work.

Part H
Did you get the same answer for Cecil’s weight in parts D and G? What can you conclude about the expressions you developed in parts D and G?

Part I
Let's try something a bit different. The four brothers know that Bruce weighs more than Alan. Write an expression to represent how much more Bruce weighs than Alan. Write the expression in terms of Dylan’s weight, d.

Part J
Dylan weighs 45 pounds. Use the expression you derived in part I to calculate how much more Bruce weighs than Alan.

Part K
Now simplify the expression (3d − 12) − (2d + 3) you wrote in part I.

Part L
Dylan weighs 45 pounds. Use the simplified expression you found in part K to recalculate how much more Bruce weighs than Alan. Did you get the same answer as in part J? If not, be sure to check your work, particularly the positive and negative signs.

A:

Dylan’s weight is d.

Twice Dylan’s weight is 2d.

Three pounds more than twice Dylan’s weight is 2d + 3.

So, the expression for Alan’s weight in terms of Dylan’s weight is 2d + 3.

B:

Dylan’s weight is d.

Three times Dylan’s weight is 3d.

Twelve less than three times Dylan’s weight is 3d − 12.

So, the expression for Bruce’s weight in terms of Dylan’s weight is 3d − 12.

C:

Cecil’s weight is the sum of Alan's weight and Bruce's weight.

Alan’s weight in terms of Dylan's weight is 2d + 3.

Bruce’s weight in terms of Dylan's weight is 3d − 12.

So, the expression for Cecil’s weight in terms of Dylan’s weight is (2d + 3) + (3d − 12).

D:

Cecil's weight is (2d + 3) + (3d − 12).

Find Cecil's weight by substituting Dylan’s weight (d = 45) into the expression above:

(2 × 45 + 3) + (3 × 45 − 12)

=  (90 + 3) + (135 − 12)

=  93 + 123

=  216.

If Dylan weighs 45 pounds, then Cecil weighs 216 pounds.

E:

Yes, the expression can be simplified by using the Associative and Commutative Properties of addition and by combining the like terms in the expression.

F:

(2d + 3) + (3d − 12)

First remove the parentheses:

2d + 3 + 3d − 12.

Then group the like terms:

2d + 3d + 3 − 12.

Finally, add and subtract the like terms:

5d − 9.

The expression (2d + 3) + (3d − 12) simplifies to 5d − 9.

G:

Cecil’s weight in terms of Dylan’s weight is 5d – 9.

Find Cecil’s weight by substituting Dylan’s weight (d = 45) into the expression above:

(5 × 45) − 9

= 225 − 9

= 216.

If Dylan weighs 45 pounds, then Cecil weighs 216 pounds.

H:

Yes, the answers are the same. The expression from part D is equivalent to the expression from part G.

I:

Bruce's weight in terms of Dylan’s weight is 3d − 12.

Alan's weight in terms of Dylan’s weight is 2d + 3.

So, the expression (3d − 12) − (2d + 3) represents how much more Bruce weighs than Alan.

J:

The expression is (3d − 12) − (2d + 3).

Evaluate the expression by substituting Dylan’s weight (d = 45) into it:

(3 × 45 − 12) − (2 × 45 + 3)

= (135 − 12) − (90 + 3)

= (123) − (93)

= 30.

Bruce weighs 30 pounds more than Alan weighs.

K:

The expression (3d − 12) − (2d + 3) simplifies to d − 15.

L:

The expression is d − 15.

Evaluate the expression by substituting Dylan’s weight (d = 45) into it:

45 − 15 = 30.

Bruce weighs 30 pounds more than Alan. This is the same answer as in part J.

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step-by-step explanation:

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