To find the smallest number of golf tees that both Angie's Sports and Lee's Gear could have sold, we start by letting \( x \) be the number of packs sold by Angie's Sports and \( y \) be the number of packs sold by Lee's Gear.
Angie's Sports sells golf tees in packs of 10, so the total number of tees sold by Angie's Sports is given by: \[ 10x \]
Lee's Gear sells golf tees in packs of 5, so the total number of tees sold by Lee's Gear is given by: \[ 5y \]
Since both shops sold the same number of golf tees, we set the two expressions equal: \[ 10x = 5y \]
To simplify this equation, we can divide both sides by 5: \[ 2x = y \]
This shows that the number of packs of tees sold by Lee's Gear (\( y \)) is twice the number of packs sold by Angie's Sports (\( x \)).
Next, we want to find the smallest number of tees sold by both shops. The total number of tees sold by both stores can be expressed as: \[ 10x \text{ (from Angie's Sports)} \quad \text{and} \quad 5y \text{ (from Lee's Gear)} \]
Since we already established that \( y = 2x \), we can substitute \( y \) into the expression for the tees sold by Lee's Gear: \[ 5y = 5(2x) = 10x \]
Thus, both expressions for total tees sold become equal to \( 10x \).
To determine the smallest number of tees each could have sold, we need to minimize \( x \). The smallest integer value for \( x \) is 1, leading to: \[ 10x = 10(1) = 10 \]
So, if \( x = 1 \):
- Angie's Sports sold \( 10 \) tees (from 1 pack).
- For \( y \), substituting \( x = 1 \) gives \( y = 2(1) = 2 \), so Lee's Gear sold: \[ 5y = 5(2) = 10 \text{ tees (from 2 packs)}. \]
Thus, both shops sold \( 10 \) tees. Therefore, the smallest number of tees each could have sold this week is: \[ \boxed{10} \]
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