Diana, Elsa, Flora and Hannah agreed to share the cost of a present equally for their mother. Hannah did not bring her money so the other 3 sisters paid for her share. Flora paid 4/5 of what Diana and Elsa paid. Diana paid $10 less than Elsa. The ratio of Diana and Elsa share is 1:4. When they reached home, Hannah returned her sisters' their share of money. Hannah returned Flora $84. (a) How much did Diana and Elsa pay altogether? (b) How much did Hannah pay Diana?

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct(when both increase or both decrease) or inverse proportional(when one increases and the other decreases, or vice versa), can be built to find the desired measures in the problem, or equations to find these measures.

Hannah returned Flora $84, and Flora paid 4/5 of what Diana and Elsa paid, hence the amount Diana and Elsa received by Hannah is given as follows:

4/5x = 84

0.8x = 84

x = 84/0.8

x = 105.

Hence Diana and Elsa paid $105 altogether.

For item b, we have that the ratio of Diana and Elsa share is 1:4, hence Diana got 1/5 of the amount paid by Hannah, hence:

1/5 x 105 = 21.

Hannah paid Diana $21.

More can be learned about proportions at https://brainly.com/question/24372153

#SPJ1