Problem with $wpdb->get_var

Can anyone shed some light on why the following doesn’t work:

function cat_has_products($id) {
	global $wpdb;

	$sql = "SELECT term_taxonomy_id from wp_term_taxonomy where term_id = $id;";
	$termtaxid = $wpdb->get_var ($wpdb->prepare( $sql) );
	echo "ID = $id <br>";
	echo "Term id = $termtaxid <br>" ;
	$catprodcount = $wpdb->get_var ($wpdb->prepare( "select count(object_id) from $wpdb->term_relationships where term_taxonomy_id = $termtaxid;") );
	echo "Count is : <br>" . $catprodcount;
	//return $catprodcount;

}

Typical output is as follows:

ID = 96
Term id =
Count is :

where 96 is the value passed to the function. The problem seems to be with the line:
$termtaxid = $wpdb->get_var ($wpdb->prepare( $sql) );
– $termtaxid is not receiving a result. However, if I hardcode the value for $id into the $sql query, $termtaxid receives the correct value.

Similarly, if I hardcode the value for $termtaxid into the last query $catprodcount = $wpdb->get_var ($wpdb->prepare( "select count(object_id) from $wpdb->term_relationships where term_taxonomy_id = $termtaxid;") );, $catprodcount receives a value.

I hope someone can help – I am tearing my (rapidly diminishing) hair out here.