Evaluating the expression 2 bags x 3 apples/bag
This Learning Outcome is vital for a happy reason – if an expression makes good literal sense (i.e., makes reference to valid real world, or equivalent scientific quantities), it can be very uniquely expressed mathematically. This Learning Outcome asserts the simplicity of math as a language.
We know that ‘2 bags x 3 apples/bag’ is the mathematical equivalent of the conversationally rational expression ‘2 bags of 3 apples each’. Let us explore if ‘2 bags x 3 apples/bag’ is valid in math.
For a change, we will start with the assumption that it is a valid expression in math (based on that it does represent something sensible in everyday conversations).
Thus, the real question is why the expression ‘2 bags x 3 apples/bag’ is valid? This question arises because we have learnt that multiplier shows the number of times something (multiplicand) is repeatedly added. In the question, the multiplier is a quantity. It is ‘2 bags’, not just a number (2).
What if the multiplicand is expressed as just ‘1 unit’?
Recall, the multiplicand is usually a multiple of something, for example, 2 cars. But what if the multiplicand is expressed as ‘1 unit.’ For example, 1 packet, 1 bowl, 1 group. This is entirely possible, and this is an interesting thought to be explored in this Learning Outcome. Consider the following multiplication expressions:
- 6 of 3 apples
- 8 of 5 pencils
- 11 of $ 100
- 6 of 40 houses
- 32 of 30 students
In all the above, multiplicands are expressed in multiple units of something. For example, 3 apples, 5 pencils, $ 100 (100 units of $ 1), 40 houses in an apartment building, 30 students in a section.
In all these examples, the multiplier needs to state the number of times these multiple quantities are added. It just needs to be a number – 6, 8, 11, 6, and 32 (in the above examples). The multiplier does not need to be anything else. For example, 32 as multiplier is just the number needed to know the total number of students in a school where all the sections have 30 students each, and there are 32 sections.
But what if the quantities mentioned as multiplicand are ‘packaged’ and each such quantity becomes a ‘1 unit’ thing, instead of being multiple units themselves. For example, 3 apples are packed together, and each 3 apples is referred to as ‘1 bag’ (and we keep the information that each bag has 3 apples). Similarly, the other quantities in the above examples can be expressed as ‘1 packet’, as under:
- 5 pencils are packed together as ‘1 packet’ (and we keep the information that each packet of the pencil has 5 pencils)
- $100 is seen as ‘1 currency note’ (and we keep the information that every currency note is $100)
- 40 houses are referred to as ‘1 apartment’ (and keep the information that there are 40 houses in 1 apartment building)
- 30 students are referred to as ‘1 section’ (and keep the information that there are 30 students in 1 section)
If the multiplicand is reduced to just ‘1 unit’ of something, as above, how does that affect the multiplication expression? By the way, most of the things around us come in group/packets and not individual things. Multiplicand expressed as group/packets is the more common reality for us.
Let us proceed with how multiplication expressions must change.
Expressions with multiplicand as ‘1 unit’.
The 5 multiplication expressions with and replaced multiplicands (as in ‘1 unit’ chosen for each of them). Here is what they will read like:
- 6 of 1 bag
- 8 of 1 packet
- 11 of 1 currency note
- 6 of 1 apartment building
- 32 of 1 section
The missing information in ‘1 unit.’
But do these expressions convey the same quantity as the original ones? For example, is ‘8 of 1 packet’ same as ‘8 of 5 pencils’, or ‘32 of 1 section’ same as ‘32 of 30 students’? Apparently, not. The expressions are missing on the details of the ‘1 unit’; that ‘1 section’ is 30 students, ‘1 apartment building’ has 40 houses. These details are an integral part of the multiplicand information; we must have these.
How can we get the missing information in the expressions? The expressions and the missing information are:
- 6 of 1 bag, but there are 3 apples/bag.
- 8 of 1 packet, but there are 5 pencils/packet.
- 11 of 1 currency note, but there are $100/currency note.
- 6 of 1 apartment building, but there are 40 houses/apartment building.
- 32 of 1 section, but there are 30 students/section.
Changing multiplier and multiplicand to integrate the missing information
There is an interesting possibility of not losing the information on quantity in each ‘1 unit’ while using ‘1 unit’ in multiplicand -
1. 6 of 1 bag, but there are 3 apples/bag
The above expression may be replaced by ‘6 bags of 3 apples/bag.’
2. 8 of 1 packet, but there are 5 pencils/packet
The above expression may be replaced by ‘8 packets of 5 pencils/packet.’
3. 11 of 1 currency note, but there are $100/currency note
The above expression may be replaced by ‘11 currency notes of $ 100/currency note.’
4. 6 of 1 apartment building, but there are 40 houses/apartment building
The above expression may be replaced by ‘6 apartment buildings of 40 houses/apartment building.’
5. 32 of 1 section, but there are 30 students/section
The above expression may be replaced by ‘32 sections of 30 students/section.’
In the above new expressions, both multiplier and multiplicand have been written differently to make a complete equivalent expression without losing information.
The new form of multiplier and multiplicand
The multiplier can be more specific than just being a number. The multiplicand can show the new ‘1 unit’ specific details used to express the quantity in multiplicand. Multiplier expresses the multiple ‘1 units’ of the multiplicand.
Thus, the information about quantity is in both - multiplier and multiplicand.
For example, in ‘6 apartment buildings of 40 houses/apartment building’, ‘6 apartment buildings’ is the new multiplier when the multiplicand gives the detail of the new ‘1 unit’ used to express the multiplication. Indeed, multiplier becomes multiplicand. But the unit of the product is expressed in multiplicand, which supports the way we define products.
Now, let us come back to our question and start with identifying the nature of the multiplicand.
The multiplicand is 3 apples per bag, i.e., it is a group of apples such that each group is a bag, and each bag has 3 apples. Thus, the multiplier will be a number with the group as its unit, i.e., bag as its unit (a bag is a group of apples).
That reality is represented by the mathematical expression: 2 bags x 3 apples/bag. It is a correct mathematical expression.
Note: The term apples/bag is not to be confused with any division expression and is to be read as apples per bag.
We know that ‘2 bags x 3 apples/bag’ is the mathematical equivalent of the conversationally sensible expression ‘2 bags of 3 apples each’. The expression ‘2 bags x 3 apples/bag’ is valid in math because 3 apples are packed together and each 3 apples is referred to as ‘1 bag’ (and we keep the information that each bag has 3 apples).
An interesting thing about multiplicand is that quantities can be expressed in two different ways, as under:
The unit of quantity is single, such as 1 person, 1 house, 1 dozen, INR 100 currency note, an apartment building, a country; every quantity is a quantity because it has been counted/measured in terms of a unit, the ‘1’
The unit of quantity is a group of multiple quantities/units, i.e., many/multiple units as a group, such as a family of 3 members, a family of 5 members, a packet of 10 ice cream bars, a packet of 25 ice cream bars.
Thus, the multiplicand can be single unit/quantity, or a group of units/quantity. Obviously, if the nature of multiplicand varies, the nature of multiplier must also vary to match with the nature of multiplicand.
There are two kinds of multiplier, corresponding to each of the two types of multiplicand:
- When the unit of the multiplicand is a single thing – the multiplier will be a number.
- When the unit of the multiplicand is a group of quantities –the multiplier will be a number with a unit. (the unit will be the group e.g., family, packet)
Thus, for the following multiplicands, the multiplier will be a number:
- Water melons
- Ice creams
And for the following multiplicands, the multiplier will be a number with a group as its unit:
- 4 dozen eggs per carton
- 12 metres per second
- 3 apples per bag
- ½ kilograms per square metre
- 7.5 rupees per half a dozen
To know if a multiplier is in the correct form, we have to evaluate the nature of multiplicand, i.e., whether it is made of a single quantity or a group of quantities.
Excerpted from the book ‘Foundations of Multiplication (Mathematics as a language)’ by Sandeep Srivastava and Saloni Srivastava