Tuesday, 22 May 2018

More Word Problems

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 1Difference 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 Draw/Working of TUV
  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
(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
        - 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
            [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
[$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)
             - 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 
           Cost = 2h x $8
                     = $16
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 =  [÷] 

               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