Less than in word problems

Fewer or Less Than?

by Owen Fourie

Which one of these two sentences is correct?

• I have fewer mistakes in my essay than you have in yours.
• I have less mistakes in my essay than you have in yours.

In the following pair, which one is correct?

• I have fewer rice on my plate than you have on yours.
• I have less rice on my plate than you have on yours.

To understand how to use words such as fewer and less correctly as you write and as you speak, you must distinguish between count nouns and mass nouns.

Count Nouns

Count nouns are words that name things that can be or should be counted as individual units.

Such words are

arrows, battleships, cars, dogs, errors, houses, pens, students, tables.

For count nouns, use the adjective fewer.

Mass Nouns

Mass nouns are words that name things that cannot be or should not be taken apart and counted as individual units.

Such words are

air, butter, clothing, dust, electricity, homework, perseverance, sand, traffic.

For mass nouns, use the adjective less.

Count Nouns or Mass Nouns?

Words such as hair, light, and noise, which can be used as mass nouns in their singular form, are used as count nouns in their plural form.

• His hair is long, but he still has less hair than she has.
• There are fewer hairs on the back of my hand than there are on yours.
• There is less light by day in the room with only one small window than in the room with the large windows.
• There are fewer lights on at night in my house than there are in yours.
• The noise of traffic disturbs me in your room, but there is less noise in my room, so I prefer to study there.
• Many different noises kept me awake at night in the city, but now that I have moved into the country, there are fewer noises to trouble me.

Distance, Time, and Money

Watch out for the exceptions relating to distance, time, and money.

• We traveled less than ten miles to the next cinema complex.
• We spent less than two hours there.
• We spent less than $20.

Although miles, hours, and dollars can be counted as separate units, the information in these sentences requires us to take them as whole amounts – a mass of distance, a mass of time, and a mass of money – so less is the appropriate adjective.

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Look again at the sentences at the beginning of this article. Now, you should be able to make the correct choice:

• I have fewer mistakes in my essay than you have in yours. (You can count the mistakes individually.)
• I have less rice on my plate than you have on yours. (It would be an absolute pain to count the individual grains of rice.)

Tactic for deciding what it is – mass or count noun?

All that you need to do is to think carefully about the object or substance described by a particular noun.

• Is it something that is obviously an individual item and there are others like it that stand as individual items? If so, it is a count noun. You can count each item and get the total of all of them together. Fewer is the adjective to use.
• Is it a substance that cannot be taken apart at all? Think of water or any liquid? If so, it is a mass noun.
• Is it something that is difficult to take apart or that would be impossible to count as individual units because there are too many little bits? Think of sand and rice. There are far too many grains of each. They are uncountable, so it is a mass noun. Less is the adjective to use.

How students learn incorrect grammar

Next time you visit a supermarket to buy only a few items, go to the checkout reserved for that purpose. Look carefully and see the sign that tells you where to go. What does it say? No doubt it will say

“20 items or less” or “Less than 20 items.”

Items are countable. The word item is a count noun, not a mass noun. The correct adjective is fewer. Do you see now how students learn incorrect grammar?

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Have you observed instances of what might be incorrect grammar in the marketplace and in advertising? Mention them here to get clarification. Your comments, observations, and questions are welcome.

Here are more articles to help you with English words, grammar, and essay writing.

Copyright © 2011 by English Essay Writing Tips www.englishessaywritingtips.com

Here is a short video on this subject that is well worth watching. The teacher presents the point with the utmost clarity:

Count Nouns and Mass Nouns (Countable and Uncountable)

Inequalities, like equations, can be translated and used to solve problems. We use an inequality for values that aren’t the same, or that can only be the same up to a certain amount.

When a problem says «at least» or «no less than,» this means the number given is the very smallest it’ll go; it can’t get any smaller. The remaining value must be something bigger. For example, if a bag of candy has at least 28 pieces, we know it has 28 or more pieces. So the number of pieces (x) is greater than or equal to 28.

x ≥ 28

The phrases «at most» and «no more than» both mean the number given is the biggest the value will ever get. If a box of drinks has no more than 15 drinks, it has 15, or 14, or 13…drinks. Notice these values are all less than or equal to 15.

x ≤ 15

Sometimes we’re lucky and the problem just uses the phrase «less than» (<) or «greater than» (>). But humans are not robots—we like to get creative about our phrasing—so don’t count on that.

Example 1: One-Step Inequalities

A bag of candy is split between us and our little brother. The bag says it has at most 28 pieces of candy in it. There are 15 candies in our bag, and x candies in our brother’s bag, and we have to make sure he doesn’t have more than us (of course). We can write an inequality expressing the number of candies in the bags.

We know that our bag has 15 candies.
We know that his bag has x candies.
We know that up to 28 candies were split between both bags, no more.
Since we know that x + 15 can’t be more than 28, it must be less than or equal to 28.

x + 15 ≤ 28

Now that we have an inequality expressing our candy, we solve for x. To do that, we subtract 15 from both sides.

x + 15 ≤ 28
x + 15 – 15 ≤ 28 – 15
x ≤ 13

Sweet, literally. Our brother has 13 or fewer pieces of candy, so no way does he have more candy than us. Until, in a moment of weakness (er, kindness) we share a few of our pieces with him. Now it’s even sweeter.

Example 2: Two-Step Inequalities

A candy store owner saw us share our candy with our brother and was so impressed he gave us a $30.00 gift card. Tax free even! We decide to buy a giant candy bar for $13.00 and then some lollipops with the remaining money. If each lollipop is $0.90, we can write and solve an inequality expressing how many lollipops we can buy.

We know that the amount we spend needs to be less than or equal to $30.00.
We know we want to buy x lollipops.
We know that each lollipop is $0.90, so the total cost of the lollipops is $0.90x.
We know the candy bar costs $13.00.
So the total cost of the candy is $13 + $0.90x, and this needs to be less than or equal to $30.00.

13 + 0.90x ≤ 30

Now we solve for x, which is the number of lollipops we can buy. We want to get x by itself, so start by subtracting 13 from both sides.

13 + 0.90x ≤ 30
13 + 0.90x – 13 ≤ 30 – 13
0.9x ≤ 17

Divide both sides by 0.9 to finish up. We recommend nabbing a calculator for this part.

We need to buy less than 18.89 lollipops, so we can buy 18 whole lollipops and have some change to spare.

Solving Word Problems in Algebra
Practice Problems

Did you read through the lesson on Inequality Word Problems? Are you ready to practice inequalities by solving these word
problems? Yes… I do know the answer by now — but — I know you can do
it! Now, I want you to prove it to yourself.

Let’s quickly recap a few things and you’ll be on your way!

Let’s keep these key words for inequalities handy:

Inequality Key Words

• at least — means greater than or equal to
• no more than — means less than or equal to
• more than — means greater than
• less than — means less than
  

Work through each problem slowly and start by identifying your
variables. Then write an inequality that represents the problem.

Once you’ve written the inequality, the hard work is done and you are ready to solve!

Don’t forget to check your answers at the end. This is definitely a habit that you want to set for yourself.

Ok… get to work!

Problem 1

Chris wants to order DVDs over the internet. Each DVD costs $15.99 and shipping for the entire order is $9.99. Chris has no more than $100 to spend.

• Write an inequality that represents Chris’ situation.
• How many DVDs can Chris order without exceeding his $100 limit?

Problem 2

Skate Land charges a $50 flat fee for birthday party rental and $5.50 per person. Joann has no more than $100 to spend on the birthday party.

• Write an inequality to represents Joann’s situation.
• How many people can Joann invite to her birthday party without exceeding her limit of $100?

Solutions

Problem 1:

Chris wants to order DVDs over the internet. Each DVD costs $15.99
and shipping for the entire order is $9.99. Chris has no more than $100
to spend.

• Write an inequality that represents Chris’ situation.

Let d = the number of DVDs purchase

15.99d + 9.99 < 100

 (This translates to 15.99 times the number of DVDs + 9.99 shipping cost must be less than or equal to 100 dollars.

• How many DVDs can Chris order without exceeding his limit?

We need to solve:
15.99d + 9.99 < 100

15.99d + 9.99 — 9.99

< 100 — 9.9915.99d<90.01
15.99    15.99d < 5.6

Since Chris cannot order 0.6 of a DVD, we must round down to 5.

Chris can order 5 DVDs without exceeding his limit of $100.

Problem 2:

Skate Land charges a $50 flat fee for birthday party rental and $5.50
per person. Joann has no more than $100 to spend on the birthday party.

• Write an inequality to represents Joann’s situation.

Let p = the number of people invited to the party

5.50p + 50 < 100

(This translates to $5.50 per person times the number of people plus $50 for the flat fee. This value must be less than or equal to $100.)

• How many people can Joann invite to her birthday party without exceeding her limit?

5.50p + 50 < 100

5.50p + 50 — 50 < 100 — 50

5.50p < 50
5.50     5.50

p < 9

Joann can invite up to 9 people without exceeding her limit of $100.

Great Job! Keep up the good work!

Now you are ready to move on to Graphing Inequalities

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