Useful techniques for Word Problems

using the units method:

Question: (When One item remains the same)
1. The ratio of the weight of the mutton to the beef is 3:4. When 57kg of the beef was sold, the ratio of the weight of the mutton to the beef becomes 7:3
(a) Find the weight of the beef before it was sold.
(b) Find the weight of the mutton.

Solution:

Before: M:B = 3:4
After: M:B = 7:3

Since the mutton remains the same, the "3" before should be the same quantity as the "7" after,
so, expressing it in equivalent ratios,

Before: M:B = 3:4 or 21:28
After: M:B = 7:3 or 21:9

Comparing before and after, B has been reduced by 19 units (28-9)
19 units = 57kg
so 1 unit = 3 kg

(a) Beef before it was sold was 28 units or 28x3 ==> 84kg
(b) Mutton (no change) weighed 21x3 ==> 63kg


Question: (When the total remains the same)
The ratio of Jessie's money to Katie's money is 3:2. If Jessie gives Katie $3.50, the ratio of their money will become 1:3
(a) How much money does Jessie have (after giving)?
(b) How much money does Katie have (after receiving)?

Before: J:K = 3:2 (total 5 units)
After: J:K = 1:3 (total 4 units)

Since J gave K money, the total money before and after remains the same, so we put it in equivalent terms. In this case the LCM of 5 and 4 gives 20.

Before: J:K = 3:2 or 12:8 (total 20 units)
After: J:K = 1:3 or 5:15 (total 20 units)

We can see that J has decreased by 7 units and K increased by 7 units
7 units = $3.50
1 unit = $0.50

(a) Jessie's money (after giving) is 5 units or $2.50
(b) Katie's money (after receiving) is 15 units or $7.50

Question: (When the difference remains the same)
The ratio of John's age to his friend's age is 2:5. In 7 years the ratio of their ages will be 3:4. What is John's age now?

Now: J:F = 2:5 (difference 3 units)
Later J:F = 3:4 (difference 1 unit)

Since the age difference is the same, we want to get the same equivalent units for the difference:

Now: J:F = 2:5 (difference 3 units)
Later J:F = 3:4 or 9:12 (difference 3 units)
We can see John and friend have increased by 7 units.
7 units = 7 years
1 unit = 1 year

John's age is now 2.

Good reference: