Unit and Value Word Problems
For the number of units and Value of unit type of word problems, the student needs to identify and the difference between the unit(U), value of each unit(V) and the total value(T)
Generally, the question requires the linking of the total value, to the unit and the $ value to find the specific answer.
T (Total Value) = U (number of unit) x V (value of each unit)
T = U x V
or for 2 objects or more
(U1 x V1) + (U2 + V2) = Total (Adding questions)
(U1 x V1) - (U2 + V2) = Difference (More/less Question)
(U1 x V1) = (U2 + V2) (Same/Equal question)
Example (PSLE)
Helen and Ivan had the same number of coins. Helen had a number of 50-cent coins and 64 20-cent coins. These coins had a mass of 1.134kg. Ivan had a number of 50-cent coins and 104 20-cent coins
(a) Who had more money in coins and by how much?
Step1: Keywords/Values and Draw/Working of TUV
<< Unit = same number, Helen:64(unit) 20-cent coins(Value), Ivan: 104 20-cent coins>>
Step 2: Link the unit and values
Difference in value:
Helen has 40 more 50-cent = 40 x $0.50 = $20
Ivan has 40 more 20-cent = 40 x $0.20 = $8
Step3 : Solve
Helen has more money. She has $20 - $8 = $12 more.
(b) Given that each 50-cent coin is 2.7g heavier than a 20-cent coin, what is the mass of Ivan's coins in kilogram?
Step 1: Difference in weight = 2.7
Step 2 : Link the unit and value
Ivan has 40 less 50-cent coin
40 x 2.7g = 108 g
= 0.108kg
Step 3: Solve
Helen's coin weight = 1.134kg
Ivan's coin weight = 1.134kg - 0.108 kg = 1.026kg
Example
Annie bought 4 times as many books as magazine. Each book cost $6 and each magazine cost $4. If she spent $60. How many books did she buy?
Step 1:[Unit: 4 times books as Magazine $value: book $6, magazine $4, total: $60]
Draw/Working:
Step2: Link the numbers
u x v = t
(4u x 6) + (u x 6) = 60
Step3: Solve by forming equation
24u + 6u = 60
30u = 60
u = 2
She bought 4 x u = 8 books
Example
A book cost $8 and a toy-set costs $4 more than the book. The childcare paid $320 for a number of books and toy-set. The ratio of the books to toy-set is 5 : 3.
(a) How many books did the childcare buy?
Step1: Keywords/Values and Draw/Working of TUV
Keywords/Value: [Ratio : 5 : 3,$value: book $12, toy $12+$8, total : $400]
Draw/Working: [For ratio]
B T
U 5 : 2
V $8 $12
T total = $320
Step2: Link the numbers
(5u x 8) + (2u x 12) = 320
Step3: Solve by forming equation
40u + 24u = 320
64u = 320
u = 320/64 = 5
The childcare bought 5 x 5 = 25 books.
Example
Betty sold 4 times as many iPads as laptops and collected a total
of $8400. Each laptop costs $325 more than an iPad. The amount collected for all the iPads sold was $3480 more than the amount collected for all the laptops sold. How many laptops did Betty sell? (17/p2/s)
Step1: Keywords/Values and
Keywords/Value: [Unit: 4 times ipad as laptop
$value: ipad sold more by $3480, total = $8400]
Draw/Working:
Step2: Link the numbers
$i + $l + 3480 = 8400
$i + $l = 8400 - 3480
= 4920
$i = $l
$l = 4920/2 = $2460
Step3: Solve by forming equation
ipad sales = $3480 + $2460 = $5940
laptop = $2460
For u unit of ipad and laptop, the values are
ipad = 5940/4 = 1486
Value of u unit of laptop = 2460
Value of u unit of ipad = 1486
Since laptop is $325 more for each laptop,
Laptop: 1486 + 325u = 2460
325u = 2460 – 1485
= 975
u = 975 / 325
= 3
Betty sold 3 laptops.
(1)Word Problems With Fixed and Variable Portions
Fixed + Variable portions word problems: One item(eg:distance) information is given, and the information is ‘tie’ to another item(eg:cost) with a Fixed and Variable Portion for calculation.
Example:
The table shows the taxi fare rate.
First Km
|
$3.20
|
Every additional 400m or less
|
$0.22
|
Kenny took a taxi from his home to his office. The distance travelled were 6.8km. How much did Kenny pay for the trip?
The first item is the distance, and the second item is the cost. The fixed and variable portions are the cost(second item) of charging(linking) for the distance (first item).
For working: Use either (a) 2-linking-lines or (b) extending the table
For working: Use either (a) 2-linking-lines or (b) extending the table
(a) 2 linking-lines for the 2 ‘items’
- Distance/Cost and F/V portion
F V
Distance(km) 1 | 5.8km
Cost($) $3.20 | ? x 0.22
(b) Extending given table
Example 1 (above Word Problem)
Method
Step1: Underline/Write keywords. Id question type (F/V)
- Distance and Portion
Dist
|
Portion
| |||
F
|
1 km
|
$3.20
|
1
|
1
|
V
|
6.8-1 = 5.8km
|
$0.22
|
5.8
|
? x 0.22
|
Example 1 (above Word Problem)
Method
Step1: Underline/Write keywords. Id question type (F/V)
- Draw 2-linking-line or extend table (as above)
- Calculate given item(distance) portion:
Remaining km = 6.8 – 1 = 5.8 km
Step2: Compute Variable Portion
Step2: Compute Variable Portion
[Change to same unit of measure]
Distance = 5.8km x1000 = 5800 m
400m portion = 5800/400
= 14.5
= 15 (because 'or less' = less than 400m' as = 1 portion)
Step3: Calculate Required Value
Step3: Calculate Required Value
[$0.22 is charged for 400 and less]
There are 15 ‘parts of journey’ = 15 x $0.22
= $3.30
Total fare = $3.20 + $3.30 = $6.50
Example 2
The bicycle rental charges are as below:
Bicycle for Rent
1st hour: $12.00
$8 per hour or part thereof from the 2nd hour onwards.
Becky rented a bicycle from 2:30pm to 5 p.m. How much did she pay?
Step1: Underline/Write keywords. Id question type (F/V)
Step1: Underline/Write keywords. Id question type (F/V)
- Draw 2-linking-lines
- Calculate given item(time) Portion
F V
Time 1 | 1½
Cost $12 | ? x $8
Total = 2:30pm to 5 pm = 2½ h, F = 1h, V = 2½ - 1 = 1½h
Step2: Compute Variable Portion
Time = 1½ h
Cost = 2h (because 'part thereof' = 'less than 1h' as = 1h)
Step3: Calculate required value
Step3: Calculate required value
Cost = 2h x $8
= $16
She paid $12 + $16 = $28.
She paid $12 + $16 = $28.
(2)Word Problems with Number Operations
Example
Alice had a long rope. She gave 6.48m of it to Andy and the rest to Danny.
(a) Andy cut his rope into equal lengths of 0.72m each. How many pieces did he cut his rope into?
(b) Danny’s rope was 3 times as long as Andy’s rope. What was the length of Alice’s rope at first.
Method
Step1: Underline Keywords and Draw information given
Andy 6.48 | Danny
For (a)
Step2: ID topic/learning and link
- cut into equal lengths… each = [÷]
- cut into equal lengths… each = [÷]
Andy 6.48 | Danny
? 0.72m [ ÷ ]
Step3: Solve and answer question
(a) 6.48/0.72 = 9
He cuts into 9 pieces.
For (b)
Step2: Id topics/learning and link
- 3 times as long = [ x ]
Andy 6.48 | Danny
? 0.72m ÷ x 3 Andy’s
Step3: Solve and answer question
Danny rope = 3 x 9
= 27 m
Alice rope = 6.48m + 27m
= 33.48m
