*Eunice, Grace and Joyce collect badges as their hobby. At first,*

Eunice gave 2/5 of her badges to Grace. Grace then gave 3/8 of what

she had to Joyce. Then Joyce gave 1/6 of what she had to Eunice. In

the end, they had 150 badges each. How many badges did Eunice

have at first?

Eunice gave 2/5 of her badges to Grace. Grace then gave 3/8 of what

she had to Joyce. Then Joyce gave 1/6 of what she had to Eunice. In

the end, they had 150 badges each. How many badges did Eunice

have at first?

Answer:

We work backwards, beginning with each of them having 150 badges in the end.

**In the end:**

Eunice: 150

Grace: 150

Joyce: 150

**Last transfer: Joyce gave 1/6 of what she had to Eunice.**

Joyce had 150 after she gave 1/6 of her badges to Eunice.

So 150 is 5/6 of her badges. She had 150 * 6 / 5 = 180 badges before, and gave 30 to Eunice.

*Before*Joyce gave her badges to Eunice, the girls had

Eunice: 150 - 30 = 120 (she had 30 fewer badges before)

Grace: 150 (unchanged)

Joyce: 150 + 30 = 180 (she had 30 more badges before)

**Second last transfer: Grace gave 3/8 of what she had to Joyce.**

Grace had 150 after she gave 3/8 of her badges to Joyce.

So 150 is 5/8 of her badges. She had 150 * 8 / 5 = 240 badges before, and gave 90 to Joyce.

Before Grace gave some of her badges to Joyce, the girls had

Eunice: 120 (unchanged)

Grace: 150 + 90 = 240 (she had 90 more badges before)

Joyce: 180 - 90 = 90 (she had 90 fewer badges before)

**First transfer: Eunice gave 2/5 of her badges to Grace.**

She had 120 badges after she gave 2/5 to Grace, so 120 is 3/5 of her badges before the transfer.

She had 120 * 5 / 3 = 200 badges before the transfer, and she gave 80 badges to Grace.

Before Eunice gave some of her badges to Grace, the girls had

Eunice: 120 + 80 = 200 (she had 80 more badges before)

Grace: 240 - 80 = 160 (she had 80 fewer badges before)

Joyce: 90 (unchanged)

*Answer:*Eunice had 200 badges at first.