Helpers are needed to prepare for a fete. Each helper can make either 2 large cakes or 35 small cakes per hour. The kitchen is available for 3 hours and the requirement is to prepare 20 large cakes and 700 small cakes. How many helpers are required to complete this work in time?

Introduction / Context:
This is a work and time type problem framed in terms of cakes made by helpers. It checks your ability to convert different items into a common unit and then determine how many workers are needed within a fixed time duration.

Given Data / Assumptions:

• Each helper can make 2 large cakes per hour, or 35 small cakes per hour.• Kitchen is available for 3 hours.• Required output: 20 large cakes and 700 small cakes.• All helpers work at the same rate and can divide time between large and small cakes as needed.

Concept / Approach:
We convert everything to a single unit of work. Since the helper rates are given as 2 large cakes or 35 small cakes per hour, we can express large cakes in terms of equivalent small cakes. Then the total required work is expressed in small cake equivalents and compared against the total capacity of h helpers over 3 hours.

Step-by-Step Solution:
Step 1: One hour of work by one helper equals 35 small cakes.Step 2: Since 2 large cakes are made in the same hour, 2 large cakes correspond to 35 small cakes.Step 3: Therefore 1 large cake corresponds to 35/2 small cake equivalents.Step 4: Required large cakes = 20, so small cake equivalents for these = 20 * (35/2) = 350.Step 5: Required small cakes = 700, so total small cake equivalents = 350 + 700 = 1050.Step 6: One helper working for 1 hour can do 35 equivalents, so in 3 hours one helper can do 3 * 35 = 105 equivalents.Step 7: Let the number of helpers be h. Total capacity in 3 hours = 105h equivalents.Step 8: We require 105h = 1050.Step 9: Therefore h = 1050 / 105 = 10.

Verification / Alternative check:
If there are 10 helpers, each can contribute 3 hours, giving 30 helper hours. One helper hour equals 35 small cake equivalents, so total work capacity is 30 * 35 = 1050 equivalents, exactly matching the required work.

Why Other Options Are Wrong:
If h = 15, 20 or 25, the total production capacity significantly exceeds the required 1050 equivalents. The question is asking for the minimum number of helpers required, so 10 is the unique correct value.

Common Pitfalls:
Some students try to assign fixed helpers to large and small cakes separately without converting to a common unit, making the problem unnecessarily complex. Others mistakenly multiply rates without including the total time factor.

Final Answer:
The fete requires 10 helpers to complete the work in 3 hours.