Formulate algebraic expressions for the following scenarios: total mass of Patrick and James, brother's age…

Question

Formulate algebraic expressions for the following scenarios: total mass of Patrick and James, brother's age in 5 years, and perimeter of a rectangular paper cutout.

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Here are the solutions to the problems.

2. Patrick's and James's total mass

Determine James's mass.
Patrick is $x$ kg. He is 5 kg heavier than James, so James is 5 kg lighter than Patrick.
James's mass $= x - 5$ kg.
Write an expression for their total mass.
Total mass = Patrick's mass + James's mass
$x + (x - 5)$
Simplify the expression.
$x + x - 5$ $2x - 5$
The total mass is $\boxed{(2x - 5)\text{ kg}}$ .

3. Brother's age in 5 years
Determine the brother's current age.
Maurine is $y$ years old. She is 6 years younger than her brother, so her brother is 6 years older than Maurine.
Brother's current age $= y + 6$ years.
Determine the brother's age in 5 years.
Brother's age in 5 years $= (y + 6) + 5$
$y + 6 + 5$ $y + 11$
The brother's age in 5 years will be $\boxed{(y + 11)\text{ years}}$ .

4. Perimeter of the rectangular paper cutout
Write the formula for the perimeter of a rectangle.
The perimeter $P$ of a rectangle with length $L$ and width $W$ is given by:
$P = 2(L + W)$
Substitute the given length and width into the formula.
Given length $L = (3y + 6)$ cm and width $W = (2y - 4)$ cm.
$P = 2((3y + 6) + (2y - 4))$
Simplify the expression inside the parentheses.
$P = 2(3y + 2y + 6 - 4)$ $P = 2(5y + 2)$
Distribute the 2.
$P = 2 \times 5y + 2 \times 2$ $P = 10y + 4$
The simplified expression for the perimeter is $\boxed{(10y + 4)\text{ cm}}$ .

5. Achieng's money

a) Write an expression to represent the total amount of money spent by Achieng.
Identify the amounts spent on each item.
Soda: sh. P
Milk: sh. 2r
Cake: sh. 45
Sum the amounts spent.
Total money spent $= P + 2r + 45$
The total money spent is $\boxed{(P + 2r + 45)\text{ sh.}}$ .

b) Write an expression to show the amount that Achieng remained with.
Identify the initial amount and the total spent.
Initial amount: sh. 500
Total spent: $(P + 2r + 45)$ sh.
Subtract the total spent from the initial amount.
Amount remained $= 500 - (P + 2r + 45)$
Distribute the negative sign.
$500 - P - 2r - 45$
Combine the constant terms.
$455 - P - 2r$
The amount Achieng remained with is $\boxed{(455 - P - 2r)\text{ sh.}}$ .

6. Wafula's milk

a) Write an expression to show the amount of milk he gets in one day.
Identify the amount of milk in the morning and afternoon.
Morning milk: $q$ litres
Afternoon milk: $q + 4$ litres
Sum the morning and afternoon milk.
Total daily milk $= q + (q + 4)$
$q + q + 4$ $2q + 4$
The amount of milk he gets in one day is $\boxed{(2q + 4)\text{ litres}}$ .

b) Write an expression to represent the total amount of milk per week.
Use the total daily milk from part (a).
Total daily milk $= (2q + 4)$ litres
Multiply the daily amount by 7 (for 7 days in a week).
Total weekly milk $= 7 \times (2q + 4)$
Distribute the 7.
$7 \times 2q + 7 \times 4$ $14q + 28$
The total amount of milk per week is $\boxed{(14q + 28)\text{ litres}}$ .

7. Simplify the following algebraic expressions:

a) $12x + 4y + 7 + 8x - 2y$
Group like terms together.
$(12x + 8x) + (4y - 2y) + 7$
Combine the like terms.
$20x + 2y + 7$
The simplified expression is $\boxed{20x + 2y + 7}$ .

b) $5(4p + 4q) + 2(3p - 8q)$
Distribute the numbers outside the parentheses.
$5 \times 4p + 5 \times 4q + 2 \times 3p + 2 \times (-8q)$ $20p + 20q + 6p - 16q$
Group like terms together.
$(20p + 6p) + (20q - 16q)$
Combine the like terms.
$26p + 4q$
The simplified expression is $\boxed{26p + 4q}$ .

Formulate algebraic expressions for the following scenarios: total mass of Patrick and James, brother's age in 5 years, and perimeter of a rectangular paper cutout.

Step-by-step solution

Determine James's mass.

Write an expression for their total mass.

Simplify the expression.

Determine the brother's current age.

Determine the brother's age in 5 years.

Write the formula for the perimeter of a rectangle.

Substitute the given length and width into the formula.

Simplify the expression inside the parentheses.

Distribute the 2.

Identify the amounts spent on each item.

Sum the amounts spent.

Identify the initial amount and the total spent.

Subtract the total spent from the initial amount.

Distribute the negative sign.

Combine the constant terms.

Identify the amount of milk in the morning and afternoon.

Sum the morning and afternoon milk.

Use the total daily milk from part (a).

Multiply the daily amount by 7 (for 7 days in a week).

Distribute the 7.

Group like terms together.

Combine the like terms.

Distribute the numbers outside the parentheses.

Group like terms together.

Combine the like terms.

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