Question(a) A potter bought clay, used $1/5$ for mugs, $5/8$ for teapots, and $7/12$ of the remainder for bowls. He had 55 oz left. How many oz of clay did he buy? (b) How many oz of clay was used on each set of items?

Studdy Solution

STEP 1

Assumptions
1. Harry used $\frac{1}{5}$ of the clay for mugs.
2. Harry used $\frac{5}{8}$ of the clay for teapots.
3. Harry used $\frac{7}{12}$ of the remaining clay for bowls.
4. Harry had 55 ounces of clay left after making all items.

STEP 2

Let's denote the total amount of clay Harry purchased as $C$ . After using $\frac{1}{5}$ of the clay for mugs, the remaining clay is $\frac{4}{5}C$ .
$Remaining\, clay\, after\, mugs = \frac{4}{5}C$

STEP 3

Next, Harry used $\frac{5}{8}$ of the remaining clay for teapots. So, the remaining clay after making teapots is $\frac{3}{8}$ of what was left after making mugs.
$Remaining\, clay\, after\, teapots = \frac{3}{8} \times \frac{4}{5}C$

STEP 4

Simplify the expression for the remaining clay after making teapots.
$Remaining\, clay\, after\, teapots = \frac{3}{8} \times \frac{4}{5}C = \frac{3 \times 4}{8 \times 5}C = \frac{3}{2 \times 5}C = \frac{3}{10}C$

STEP 5

Harry then used $\frac{7}{12}$ of the remaining clay for bowls. The remaining clay after making bowls is $\frac{5}{12}$ of what was left after making teapots.
$Remaining\, clay\, after\, bowls = \frac{5}{12} \times \frac{3}{10}C$

STEP 6

Simplify the expression for the remaining clay after making bowls.
$Remaining\, clay\, after\, bowls = \frac{5}{12} \times \frac{3}{10}C = \frac{5 \times 3}{12 \times 10}C = \frac{1 \times 1}{4 \times 4}C = \frac{1}{16}C$

STEP 7

According to the problem, the remaining clay after making all items is 55 ounces. We can set up the equation with the remaining clay equal to 55 ounces.
$\frac{1}{16}C = 55\, ounces$

STEP 8

To find the total amount of clay $C$ , we multiply both sides of the equation by 16.
$C = 55 \times 16$

STEP 9

Calculate the total amount of clay $C$ that Harry purchased.
$C = 55 \times 16 = 880\, ounces$

STEP 10

Now that we know the total amount of clay, we can find out how much was used for mugs. Harry used $\frac{1}{5}$ of the total clay for mugs.
$Clay\, for\, mugs = \frac{1}{5}C$

STEP 11

Calculate the amount of clay used for mugs.
$Clay\, for\, mugs = \frac{1}{5} \times 880 = 176\, ounces$

STEP 12

Next, we find out how much clay was used for teapots. Harry used $\frac{5}{8}$ of the remaining clay after mugs for teapots.
$Clay\, for\, teapots = \frac{5}{8} \times \frac{4}{5}C$

STEP 13

Simplify and calculate the amount of clay used for teapots.
$Clay\, for\, teapots = \frac{5}{8} \times \frac{4}{5} \times 880 = \frac{5 \times 4}{8} \times 176 = \frac{20}{8} \times 176 = \frac{5}{2} \times 176 = 2.5 \times 176 = 440\, ounces$

STEP 14

Finally, we find out how much clay was used for bowls. Harry used $\frac{7}{12}$ of the remaining clay after teapots for bowls.
$Clay\, for\, bowls = \frac{7}{12} \times \frac{3}{10}C$

STEP 15

Simplify and calculate the amount of clay used for bowls.
$Clay\, for\, bowls = \frac{7}{12} \times \frac{3}{10} \times 880 = \frac{7 \times 3}{12 \times 10} \times 880 = \frac{21}{120} \times 880 = \frac{7}{40} \times 880 = 7 \times 22 = 154\, ounces$
(a) Harry purchased 880 ounces of clay. (b) Harry used 176 ounces of clay for mugs, 440 ounces for teapots, and 154 ounces for bowls.

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