Hypothesis testins as a tool for GMAT math problem solving

Certain problems require you to formulate a hypothesis and verify. The relationships between relevant variables which are yet unknown but promise to offer solution in full or in part forms the basis of this method. Such formulations are tested for validity and accepted or rejected. More than one hypothesis can be formulated in a problem context. These hypothesis have to be examined and reformulated.

Errors which occur during hypothesis testing
(1) Overlooking certain data
(2) Overemphasizing data which give positive conclusions while failing to give sufficient importance on data which falsifies information.

The following sum is a tough mathematical problem where skill of hypothesizing information and testing it comes forth.

Fifteen years back Mrs. John had only three daughters Rachael, Annie, Mary and their combined age was half of hers. During the next 5 years, Thomas was born. At that time Mrs. John’s age equaled the total of all her children’s ages. After some years George was born and then Rachael was as old as Mary and Thomas together. And now, the combined age of all the children is double Mrs. John’s age, which is only equal to that of Rachael and Annie together. Rachael’s age is also equal to the combined age of the two sons’.

What is Mrs. John’s age?

1. 39        2. 34          3. 29         4.24

This sum takes a long time if you solve by writing equations.

It can be solved faster by hypothesizing a data and testing the hypothesis wrt to other conditions.

First let us interpret the problem carefully and diagram it.

PROBLEM ANALYSIS

Information which is direct
• 15 years back Mrs. John had only three children Rachael, Mary and Annie. Mrs. John’s age was double the sum of the ages of her children.
• Sometime between 15 and ten years back, Thomas was born. At that time Mrs. John’s age was equal to the sum of the 3 children
• Between 10 years back and present time, George was born. At that time Rachael was as old as Mary and Thomas together.
• At present the combined age of all the children is double Mrs. John’s age. Mrs. John’s age is equal to the sum of Rachael and Annie. Rachael’s age is equal to the sum of George and Thomas

Implicit information
• All the ages are whole positive number, there are no fractions.
• Thomas’s age must be less than 15 and near 15. As 15 years back Mrs. John had only 3 children
• The last statement states that Rachael’s age + Annie’s age = Mary’s age + Thomas’s age + George’s age.
Rachael’s age = George’s age + Thomas’s age and Mrs. John = Rachael’s age + Annie’s age

PROBLEM CONVERSION

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-      George

T+G+A    -   Present  - T+G    -A      -A      -T         -G

Where T is Thomas age, A is Annies age and G is Georges age
We can conclude that Annie and Mary were twins


We will assume data from the questions
Let us hypothesize that Thomas’s age is 12. (9 is far away from 15).
{If we don’t get the answer using T =12 we can conclude that T = 9. Other options are wrong.}

Lets verify our hypothesis.

Rachael must be the eldest daughter . Let us assume that Rachael age must be 21 other options are close to 15 and as she is the eldest we will assume the biggest number
So George’s age must be 9 ( question 4 seems to be satisfied)

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

                    15 yrs back-6     -

                     12 yrs back-9    -                         -0(Thomas born)

                      10 yrs back-11 -                        - 2

                      9 yrs back - 12 -                        -3     -0(Georges born)

                      5 yrs back  - 16-                        7       - 4

T+G+A              Present  -21 -     A-   A-       12      -   9

Using the info: Between 10 years back and present time, George was born. At that time Rachael was as old as Mary and Thomas together.
It can be concluded that Mary was 9 as 12 = Mary’s age + 3
Hence Annie’s age was also 39

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

                    9 yrs back- 12     -  9      -9     -  3        - 0(Georges born)

Filling our table
All the children’s age 9 yrs back can be calculated.
Their present ages can also be calculated including Mrs. John age

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

39            -  Present   - 21        -18    - 18   -   12      -  9

Mrs. John -  Time line- Rachael-Annie-Mary-Thomas-George

24          -  15 yrs back -  6     -      3  -   3

               -12 yrs back  -  9      -    6   -   6 -  0(Thomas born)

               - 10 yrs back - 11                        -  2

                - 9 yrs back  - 12     -   9    -   9  -  3       -0(Georges born)

               -  5 yrs back   -16                         - 7        -  4

      39       -Present     - 21       -  18    -  18  -12     -   9

Now the table can be completed and the all the answers can be calculated

What is Mrs. John’s age? - 39
What is the age of the eldest daughter? - 21
What is the age of the eldest son, Thomas ? -12
What is the age of the youngest child? - 9



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