Question

write an algebraic expression for each of the given quantities.
four friends have different numbers of cows on their farms. jayden has 3 times as many cows as harper. mateo has half as many cows as jayden. sofia has 2 less than 4 times as many cows as harper. together, the four friends have 90 less than 5 times the number of cows jayden has.
determine the number of cows each friend has. let ( x ) represent the number of cows harper has.
→ enter the answer in each space provided. use numbers instead of words.

answer the question by writing and solving an equation.
→ enter the answer in each space provided. use numbers instead of words.
equation
harper has (square) cows, jayden has (square) cows, mateo has (square) cows, and sofia has (square) cows.

Explanation:

Step1: Define variables for each friend

Let $x$ = number of cows Harper has.
Jayden: $3x$
Mateo: $\frac{1}{2}(3x) = \frac{3x}{2}$
Sofia: $4x - 2$

Step2: Set up total cows equation

Total cows: $x + 3x + \frac{3x}{2} + 4x - 2 = 5(3x) - 90$

Step3: Simplify left side

Combine like terms:
$x + 3x + 4x + \frac{3x}{2} - 2 = 8x + \frac{3x}{2} - 2 = \frac{16x + 3x}{2} - 2 = \frac{19x}{2} - 2$

Step4: Simplify right side

$15x - 90$

Step5: Solve for $x$

$\frac{19x}{2} - 2 = 15x - 90$
Multiply all terms by 2:
$19x - 4 = 30x - 180$
Rearrange terms:
$180 - 4 = 30x - 19x$
$176 = 11x$
$x = \frac{176}{11} = 16$

Step6: Calculate each friend's cows

Harper: $x = 16$
Jayden: $3x = 3(16) = 48$
Mateo: $\frac{3x}{2} = \frac{3(16)}{2} = 24$
Sofia: $4x - 2 = 4(16) - 2 = 62$

Answer:

Expressions:

Cows Jayden has: $3x$
Cows Mateo has: $\frac{3x}{2}$
Cows Sofia has: $4x-2$

Equation:

$x + 3x + \frac{3x}{2} + 4x - 2 = 15x - 90$

Final Counts:

Harper has 16 cows, Jayden has 48 cows, Mateo has 24 cows, and Sofia has 62 cows.